Talk slides

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a cellular basis. The proof involves Dynkin elements (nested commutators) of the free algebra and their interactions with the B-basis.

We generalize Reiner-Saliola-Welker's well-known but mysterious family of k-random-to-random shuffles from Markov chains on symmetric groups to Markov chains on the Type-A Iwahori–Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in q with non-negative integer coefficients. Our work generalizes work of Reiner–Saliola–Welker and Lafrenière for the symmetric group, and simplifies all known proofs in this case.

We discuss the hook length formula and some related results.

We generalize Reiner-Saliola-Welker's well-known but mysterious family of k-random-to-random shuffles from Markov chains on symmetric groups to Markov chains on the Type-A Iwahori–Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in q with non-negative integer coefficients. Our work generalizes work of Reiner–Saliola–Welker and Lafrenière for the symmetric group, and simplifies all known proofs in this case.

We generalize Reiner-Saliola-Welker's well-known but mysterious family of k-random-to-random shuffles from Markov chains on symmetric groups to Markov chains on the Type-A Iwahori–Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in q with non-negative integer coefficients. Our work generalizes work of Reiner–Saliola–Welker and Lafrenière for the symmetric group, and simplifies all known proofs in this case.

Motivated by a question and some enumerative conjectures of Richard Stanley, we explore the equivalence classes of words in the Weyl algebra, k < D,U | DU - UD = 1 >. We show that each class is generated by the swapping of adjacent balanced subwords, i.e., those which have the same number of D's as U's, and give several other characterizations, as well as a linear-time algorithm for equivalence checking. Armed with this, we deduce several enumerative results about such equivalence classes and their sizes. We extend these results to the class of c-Dyck words, where every prefix has at least c times as many U's as D's. We also connect these results to previous work on bond percolation and rook theory, and generalize them to some other algebras.

Let 𝒜 be the group algebra k[S_n] of the n-th symmetric group S_n over a commutative ring k. For any two subsets A and B of [n] = {1, 2, ..., n}, we define the elements

∇_B,A := sum of all permutations w ∈ S_n satisfying w(A) = B

and

˜∇_B,A := sum of all permutations w ∈ S_n satisfying w(A) ⊆ B

of 𝒜. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). We express the product ∇_D,C ∇_B,A as a Z-linear combination of ∇_U,V's.

More generally, for any two set compositions (i.e., ordered set partitions) A and B of [n], we define ∇_B,A to be the sum of all permutations w ∈ S_n that send each block of A to the corresponding block of B. This generalizes ∇_B,A. The factorization property of minimal polynomials does not extend to the ∇_B,A, but we describe the ideal spanned by the ∇_B,A and a further ideal complementary to it. These two ideals have a "mutually annihilative" relationship, are free as k-modules, and appear as annihilators of tensor product S_n-representations; they are also closely related to Murphy's cellular bases, Specht modules, pattern-avoiding permutations and even some algebras appearing in quantum information theory.

Let D = (V, A) be a digraph with n vertices, where each arc a ∈ A is a pair (u, v) of two vertices. We study the Redei--Berge symmetric function U_D, defined as the quasisymmetric function ∑ L_{Des(w, D), n} ∈ QSym. Here, the sum ranges over all lists w = (w₁, w₂, ..., w_n) that contain each vertex of D exactly once, and the corresponding addend is L_{Des(w, D), n} := ∑ x_i1 x_i2 ... x_in summed over all n-tuples (i₁ ≤ i₂ ≤ ... ≤ i_n) of positive integers that satisfy i_p < i_p+1 for each p satisfying (w_p, w_p+1) ∈ A (an instance of Gessel's fundamental quasisymmetric functions).

While U_D is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express U_D in terms of the power-sum symmetric functions. We show that U_D is always p-integral, and furthermore is p-positive whenever D has no 2-cycles. When D is a tournament, U_D can be written as a polynomial in p₁, 2p₃, 2p₅, 2p₇, ... with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-4 refinement of Redei's theorem.

We explore several generalizations of Whitney's theorem -- a classical formula for the chromatic polynomial of a graph. Following Stanley, we replace the chromatic polynomial by the chromatic symmetric function. Following Dohmen and Trinks, we exclude not all but only an (arbitrarily selected) set of broken circuits, or even weigh these broken circuits with weight monomials instead of excluding them. Following Crew and Spirkl, we put weights on the vertices of the graph. Following Gebhard and Sagan, we lift the chromatic symmetric function to noncommuting variables. In addition, we replace the graph by an "ambigraph", an apparently new concept that includes both hypergraphs and multigraphs as particular cases. We show that Whitney's formula endures all these generalizations, and a fairly simple sign-reversing involution can be used to prove it in each setting. Furthermore, if we restrict ourselves to the chromatic polynomial, then the graph can be replaced by a matroid. We discuss an application to transitive digraphs (i.e., posets), and reprove an alternating-sum identity by Dahlberg and van Willigenburg.

If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H → H ⊗ QSym is defined. This homomorphism generalizes the "internal comultiplication" on QSym, and extends what Hazewinkel (in Section 18.24 of his "Witt vectors") calls the Bernstein homomorphism.

We construct this homomorphism with the help of the universal property of QSym as a combinatorial Hopf algebra (a well-known result by Aguiar, Bergeron and Sottile) and extension of scalars (the commutativity of H allows us to consider, for example, H ⊗ QSym as an H-Hopf algebra, and this change of viewpoint significantly extends the reach of the universal property).

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained.

Consider labellings of a finite poset P by |P| + 2 elements of a ring K: one label associated with each poset element and two constant labels for the added top and bottom elements. Birational rowmotion is a partial map on such labellings. It was originally defined by Einstein and Propp for K = R as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope O(P) := {order-preserving f : P → [0,1]}. The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of P, which correspond to the vertices of O(P). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing K to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.

In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when P is a product of two chains) for K a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.

We study an infinite family of shuffling operators on the symmetric group S_n, which includes the well-studied top-to-random shuffle. The general shuffling scheme consists of removing one card at a time from the deck (according to some probability distribution) and re-inserting it at a position chosen uniformly at random among the positions below. Rewritten in terms of the group algebra R[S_n], our shuffle corresponds to right multiplication by a linear combination of the elements

t_ℓ := cyc_ℓ + cyc_ℓ,ℓ+1 + cyc_{ℓ,ℓ+1,ℓ+2} + ... + cyc_{ℓ,ℓ+1,...,n} ∈ R[S_n]

for all ℓ ∈ {1, 2, ..., n} (where cyc_{i1, i2, ..., ip} denotes the permutation in S_n that cycles through i₁, i₂, ..., i_p).

We compute the eigenvalues of these shuffling operators and of all their linear combinations. In particular, we show that the eigenvalues of right multiplication by a linear combination λ₁t₁ + λ₂t₂ + ... + λ_nt_n (with λ₁, λ₂, ..., λ_n being reals) are the numbers λ₁m_I,1 + λ₂m_I,2 + ... + λ_nm_I,n, where I ranges over the lacunar subsets of {1, 2, ..., n-1} (i.e., over the subsets that contain no two consecutive integers), and where m_I,ℓ denotes the distance from ℓ to the next-higher element of I (which element is understood to be ℓ itself if ℓ ∈ I, and to be n+1 if ℓ > max I). We compute the multiplicities of these eigenvalues and show that if they are all distinct, the shuffling operator is diagonalizable. To this purpose, we show that the operators of right multiplication by t₁, t₂, ..., t_n on R[S_n] are simultaneously triangularizable, and in fact there is a combinatorially defined basis (the "descent-destroying basis", as we call it) of R[S_n] in which they are represented by upper-triangular matrices. The results stated here over R for convenience are actually stated and proved over an arbitrary commutative ring. We finish by describing a strong stationary time for the random-to-below shuffle, which is the shuffle in which the card that moves below is selected uniformly at random, and we give the waiting time for this event to happen.

Student project mentored via the Yulia's Dream program 2022-2023. (Proof found by the students, writing mostly by me.)

The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak and Postnikov, replacing the number by a (rational) generating function, and one by Naruse, which generalizes the setting from a partition to a skew partition. Both generalizations appear to lie significantly deeper, with no simple proofs known. We combine them into a generating-function identity for skew partitions, and prove it in a fairly elementary way using recursion, determinants and simple combinatorics.

Let D = (V, A) be a digraph with n vertices, where each arc a ∈ A is a pair (u, v) of two vertices. We study the Redei--Berge symmetric function U_D, defined as the quasisymmetric function ∑ L_{Des(w, D), n} ∈ QSym. Here, the sum ranges over all lists w = (w₁, w₂, ..., w_n) that contain each vertex of D exactly once, and the corresponding addend is L_{Des(w, D), n} := ∑ x_i1 x_i2 ... x_in summed over all n-tuples (i₁ ≤ i₂ ≤ ... ≤ i_n) of positive integers that satisfy i_p < i_p+1 for each p satisfying (w_p, w_p+1) ∈ A (an instance of Gessel's fundamental quasisymmetric functions).

While U_D is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express U_D in terms of the power-sum symmetric functions. We show that U_D is always p-integral, and furthermore is p-positive whenever D has no 2-cycles. When D is a tournament, U_D can be written as a polynomial in p₁, 2p₃, 2p₅, 2p₇, ... with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-4 refinement of Redei's theorem.

Let D = (V, A) be a digraph with n vertices, where each arc a ∈ A is a pair (u, v) of two vertices. We study the Redei--Berge symmetric function U_D, defined as the quasisymmetric function ∑ L_{Des(w, D), n} ∈ QSym. Here, the sum ranges over all lists w = (w₁, w₂, ..., w_n) that contain each vertex of D exactly once, and the corresponding addend is L_{Des(w, D), n} := ∑ x_i1 x_i2 ... x_in summed over all n-tuples (i₁ ≤ i₂ ≤ ... ≤ i_n) of positive integers that satisfy i_p < i_p+1 for each p satisfying (w_p, w_p+1) ∈ A (an instance of Gessel's fundamental quasisymmetric functions).

While U_D is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express U_D in terms of the power-sum symmetric functions. We show that U_D is always p-integral, and furthermore is p-positive whenever D has no 2-cycles. When D is a tournament, U_D can be written as a polynomial in p₁, 2p₃, 2p₅, 2p₇, ... with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-4 refinement of Redei's theorem.

Let D = (V, A) be a digraph with n vertices, where each arc a ∈ A is a pair (u, v) of two vertices. We study the Redei--Berge symmetric function U_D, defined as the quasisymmetric function ∑ L_{Des(w, D), n} ∈ QSym. Here, the sum ranges over all lists w = (w₁, w₂, ..., w_n) that contain each vertex of D exactly once, and the corresponding addend is L_{Des(w, D), n} := ∑ x_i1 x_i2 ... x_in summed over all n-tuples (i₁ ≤ i₂ ≤ ... ≤ i_n) of positive integers that satisfy i_p < i_p+1 for each p satisfying (w_p, w_p+1) ∈ A (an instance of Gessel's fundamental quasisymmetric functions).

While U_D is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express U_D in terms of the power-sum symmetric functions. We show that U_D is always p-integral, and furthermore is p-positive whenever D has no 2-cycles. When D is a tournament, U_D can be written as a polynomial in p₁, 2p₃, 2p₅, 2p₇, ... with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-4 refinement of Redei's theorem.

This is an algebraic approach to the Bhargava greedoid introduced previously in a joint paper with Fedor Petrov. Here I show that any Bhargava greedoid is a Gaussian elimination greedoid (a greedoidal analogue of a representable matroid).

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained.

Consider labellings of a finite poset P by |P| + 2 elements of a ring K: one label associated with each poset element and two constant labels for the added top and bottom elements. Birational rowmotion is a partial map on such labellings. It was originally defined by Einstein and Propp for K = R as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope O(P) := {order-preserving f : P → [0,1]}. The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of P, which correspond to the vertices of O(P). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing K to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.

In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when P is a product of two chains) for K a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.

We study an infinite family of shuffling operators on the symmetric group S_n, which includes the well-studied top-to-random shuffle. The general shuffling scheme consists of removing one card at a time from the deck (according to some probability distribution) and re-inserting it at a position chosen uniformly at random among the positions below. Rewritten in terms of the group algebra R[S_n], our shuffle corresponds to right multiplication by a linear combination of the elements

t_ℓ := cyc_ℓ + cyc_ℓ,ℓ+1 + cyc_{ℓ,ℓ+1,ℓ+2} + ... + cyc_{ℓ,ℓ+1,...,n} ∈ R[S_n]

for all ℓ ∈ {1, 2, ..., n} (where cyc_{i1, i2, ..., ip} denotes the permutation in S_n that cycles through i₁, i₂, ..., i_p).

We compute the eigenvalues of these shuffling operators and of all their linear combinations. In particular, we show that the eigenvalues of right multiplication by a linear combination λ₁t₁ + λ₂t₂ + ... + λ_nt_n (with λ₁, λ₂, ..., λ_n being reals) are the numbers λ₁m_I,1 + λ₂m_I,2 + ... + λ_nm_I,n, where I ranges over the lacunar subsets of {1, 2, ..., n-1} (i.e., over the subsets that contain no two consecutive integers), and where m_I,ℓ denotes the distance from ℓ to the next-higher element of I (which element is understood to be ℓ itself if ℓ ∈ I, and to be n+1 if ℓ > max I). We compute the multiplicities of these eigenvalues and show that if they are all distinct, the shuffling operator is diagonalizable. To this purpose, we show that the operators of right multiplication by t₁, t₂, ..., t_n on R[S_n] are simultaneously triangularizable, and in fact there is a combinatorially defined basis (the "descent-destroying basis", as we call it) of R[S_n] in which they are represented by upper-triangular matrices. The results stated here over R for convenience are actually stated and proved over an arbitrary commutative ring. We finish by describing a strong stationary time for the random-to-below shuffle, which is the shuffle in which the card that moves below is selected uniformly at random, and we give the waiting time for this event to happen.

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained.

Consider labellings of a finite poset P by |P| + 2 elements of a ring K: one label associated with each poset element and two constant labels for the added top and bottom elements. Birational rowmotion is a partial map on such labellings. It was originally defined by Einstein and Propp for K = R as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope O(P) := {order-preserving f : P → [0,1]}. The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of P, which correspond to the vertices of O(P). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing K to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.

In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when P is a product of two chains) for K a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained.

Consider labellings of a finite poset P by |P| + 2 elements of a ring K: one label associated with each poset element and two constant labels for the added top and bottom elements. Birational rowmotion is a partial map on such labellings. It was originally defined by Einstein and Propp for K = R as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope O(P) := {order-preserving f : P → [0,1]}. The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of P, which correspond to the vertices of O(P). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing K to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.

In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when P is a product of two chains) for K a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.

Fix a finite undirected graph G and a vertex v of G. Let E be the set of edges of G; assume that E ≠ ∅. We call a subset F of E pandemic if each edge of G has at least one endpoint that can be connected to v by an F-path (i.e., a path using edges from F only). In 1984, Elser showed that the sum of (-1)^|F| over all pandemic subsets F of E is 0. We give a simpler proof and discuss variants and generalizations.

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

This is an algebraic approach to the Bhargava greedoid introduced previously in a joint paper with Fedor Petrov. Here I show that any Bhargava greedoid is a Gaussian elimination greedoid (a greedoidal analogue of a representable matroid).

This is an algebraic approach to the Bhargava greedoid introduced previously in a joint paper with Fedor Petrov. Here I show that any Bhargava greedoid is a Gaussian elimination greedoid (a greedoidal analogue of a representable matroid).

We prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre. The proof relies on an (apparently novel) birational involution defined over any semiring.

We prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre. The proof relies on an (apparently novel) birational involution defined over any semiring.

For any positive integer k and nonnegative integer m, we consider the symmetric function G(k, m) defined as the sum of all monomials of degree m that involve only exponents smaller than k. We call G(k, m) a Petrie symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to {0, 1, -1} by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form G(k, m) · s_μ in the Schur basis whenever μ is a partition; all coefficients in this expansion belong to {0, 1, -1}. We also show that G(k, 1), G(k, 2), G(k, 3), ... form an algebraically independent generating set for the symmetric functions when 1 - k is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of G(k, 2k-1) in the Schur basis.

For any positive integer k and nonnegative integer m, we consider the symmetric function G(k, m) defined as the sum of all monomials of degree m that involve only exponents smaller than k. We call G(k, m) a Petrie symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to {0, 1, -1} by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form G(k, m) · s_μ in the Schur basis whenever μ is a partition; all coefficients in this expansion belong to {0, 1, -1}. We also show that G(k, 1), G(k, 2), G(k, 3), ... form an algebraically independent generating set for the symmetric functions when 1 - k is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of G(k, 2k-1) in the Schur basis.

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

Fix integers n ≥ k ≥ 0. Consider the ring S of symmetric polynomials in k variables over an arbitrary base ring k. Fix k scalars a₁, a₂, ..., a_k ∈ k.

Let I be the ideal of S generated by h_n-k+1 - a₁, h_n-k+2 - a₂, ..., h_n - a_k (where h_i stands for the i-th complete homogeneous symmetric polynomial).

The quotient ring S/I generalizes both the usual and the quantum cohomology of the Grassmannian.

We show that S/I has a k-module basis consisting of (residue classes of) Schur polynomials fitting into a k × (n-k)-rectangle; and that its multiplicative structure constants satisfy the same S₃-symmetry as those of the Grassmannian cohomology. We furthermore find an analogue of the Pieri rule for complete homogeneous symmetric polynomials and a few other formulas.

We also study the quotient of the whole polynomial ring (not just the symmetric polynomials) by the ideal generated by the same k polynomials as I.

This is a continuation of the paper "Shuffle-compatible permutation statistics" by Ira M. Gessel and Yan Zhuang. We show that the exterior peak set is a shuffle-compatible permutation statistic (as conjectured by Gessel and Zhuang), using a notion of "Z-enriched (P, γ)-partitions" that generalizes the concepts of "P-partitions", "enriched P-partitions" and "left enriched P-partitions". Furthermore, we introduce the notion of "LR-shuffle-compatibility", which is a property stronger than shuffle-compatibility, and which we also verify for the permutation statistics Des, des, Lpk and Epk (but not maj, Rpk and Pk). Furthermore, we describe the kernel of the homomorphism from QSym to the shuffle algebra of the exterior peak set statistic (by finding two generating sets for this kernel), and we relate LR-shuffle-compatibility to dendriform algebra quotients of QSym in the same way as shuffle-compatibility itself relates to algebra quotients of QSym. We pose various questions about these concepts.

This is a continuation of the paper "Shuffle-compatible permutation statistics" by Ira M. Gessel and Yan Zhuang. We show that the exterior peak set is a shuffle-compatible permutation statistic (as conjectured by Gessel and Zhuang), using a notion of "Z-enriched (P, γ)-partitions" that generalizes the concepts of "P-partitions", "enriched P-partitions" and "left enriched P-partitions". Furthermore, we introduce the notion of "LR-shuffle-compatibility", which is a property stronger than shuffle-compatibility, and which we also verify for the permutation statistics Des, des, Lpk and Epk (but not maj, Rpk and Pk). Furthermore, we describe the kernel of the homomorphism from QSym to the shuffle algebra of the exterior peak set statistic (by finding two generating sets for this kernel), and we relate LR-shuffle-compatibility to dendriform algebra quotients of QSym in the same way as shuffle-compatibility itself relates to algebra quotients of QSym. We pose various questions about these concepts.

Using the description of multiline queues as functions on words, we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues. We show that the spectral weight of a word is invariant under a natural action of the symmetric group, giving a proof of the commutativity conjecture of Arita, Ayyer, Mallick, and Prolhac. We give a determinant formula for the spectral weight of a word, which gives a proof of a conjecture of Aas and Linusson.

Using the description of multiline queues as functions on words, we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues. We show that the spectral weight of a word is invariant under a natural action of the symmetric group, giving a proof of the commutativity conjecture of Arita, Ayyer, Mallick, and Prolhac. We give a determinant formula for the spectral weight of a word, which gives a proof of a conjecture of the first author and Linusson.

This is a continuation of the paper "Shuffle-compatible permutation statistics" by Ira M. Gessel and Yan Zhuang. We show that the exterior peak set is a shuffle-compatible permutation statistic (as conjectured by Gessel and Zhuang), using a notion of "Z-enriched (P, γ)-partitions" that generalizes the concepts of "P-partitions", "enriched P-partitions" and "left enriched P-partitions". Furthermore, we introduce the notion of "LR-shuffle-compatibility", which is a property stronger than shuffle-compatibility, and which we also verify for the permutation statistics Des, des, Lpk and Epk (but not maj, Rpk and Pk). Furthermore, we describe the kernel of the homomorphism from QSym to the shuffle algebra of the exterior peak set statistic (by finding two generating sets for this kernel), and we relate LR-shuffle-compatibility to dendriform algebra quotients of QSym in the same way as shuffle-compatibility itself relates to algebra quotients of QSym. We pose various questions about these concepts.

This is a continuation of the paper "Shuffle-compatible permutation statistics" by Ira M. Gessel and Yan Zhuang. We show that the exterior peak set is a shuffle-compatible permutation statistic (as conjectured by Gessel and Zhuang), using a notion of "Z-enriched (P, γ)-partitions" that generalizes the concepts of "P-partitions", "enriched P-partitions" and "left enriched P-partitions". Furthermore, we introduce the notion of "LR-shuffle-compatibility", which is a property stronger than shuffle-compatibility, and which we also verify for the permutation statistics Des, des, Lpk and Epk (but not maj, Rpk and Pk). Furthermore, we describe the kernel of the homomorphism from QSym to the shuffle algebra of the exterior peak set statistic (by finding two generating sets for this kernel), and we relate LR-shuffle-compatibility to dendriform algebra quotients of QSym in the same way as shuffle-compatibility itself relates to algebra quotients of QSym. We pose various questions about these concepts.

This is an exposition of Newman's diamond lemma, with (an outline of) a constructive proof and a few applications (including a glimpse at Gröbner bases). The target audience are students somewhat familiar with combinatorics.

Here we consider an invariant of a module over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

This is an attempt to construct an object which relates to the ring of symmetric functions in the same way as the polynomial ring F_q[T] (over a finite field) relates to the ring of integers. So, for example, while the ring of symmetric functions has bases indexed by partitions (~ conjugacy classes of permutations), this mythical object would have bases indexed by "function-field partitions" (~ conjugacy classes of matrices in GL_n(F_q)).

I have not been able to say much about this object so far; the construction I have is rather indirect and the combinatorial meaning is yet to be found. But along the way, I found a nice (I believe) F_q[T]-analogue of Witt vectors in which the Carlitz action replaces exponentiation.

This is far from finished, and most proofs have yet to be written up.

This is an attempt to construct an object which relates to the ring of symmetric functions in the same way as the polynomial ring F_q[T] (over a finite field) relates to the ring of integers. So, for example, while the ring of symmetric functions has bases indexed by partitions (~ conjugacy classes of permutations), this mythical object would have bases indexed by "function-field partitions" (~ conjugacy classes of matrices in GL_n(F_q)).

I have not been able to say much about this object so far; the construction I have is rather indirect and the combinatorial meaning is yet to be found. But along the way, I found a nice (I believe) F_q[T]-analogue of Witt vectors in which the Carlitz action replaces exponentiation.

This is far from finished, and most proofs have yet to be written up.

The reduced expressions for a given element w of a Coxeter group (W, S) can be regarded as the vertices of a directed graph R(w); its arcs correspond to the braid moves. Specifically, an arc goes from a reduced expression a to a reduced expression b when b is obtained from a by replacing a contiguous subword of the form stst... (for some distinct s, t ∈ S) by tsts... (where both subwords have length m_{s, t}, the order of st ∈ W). We prove a strong bipartiteness-type result for this graph R(w): Not only does every cycle of R(w) have even length; actually, the arcs of R(w) can be colored (with colors corresponding to the type of braid moves used), and to every color c corresponds an "opposite" color c^op (corresponding to the reverses of the braid moves with color c), and for any color c, the number of arcs in any given cycle of R(w) having color in {c, c^op} is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbé.

Slides for a talk on quasisymmetric functions and Hopf algebras. Chapter 1 was presented at the Brandeis Combinatorics Seminar on 5 April 2016; all three chapters were presented at MIT on 11 April 2016.

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.

A number of authors have studied a natural operation (under various names) on the order ideals (equivalently, antichains) of a finite poset, here called rowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes (instead of acting on order ideals, the operation here acts on points inside the order polytope of the poset), then via detropicalization to the birational setting (here, the operation acts -- more or less -- on maps from the poset to an arbitrary field).

In the latter setting, it is no longer a priori clear even that birational rowmotion has finite order, and for many posets the order is indeed infinite. However, we show that, for the poset P = [p] × [q] (product of two chains), birational rowmotion has the same order, p+q, as ordinary rowmotion. We also show that birational (hence also ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture, of which our result can be considered an analogue.

The proofs are at most sketched in the above abstract, while the main paper offers more detail.

A number of authors have studied a natural operation (under various names) on the order ideals (equivalently, antichains) of a finite poset, here called rowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes (instead of acting on order ideals, the operation here acts on points inside the order polytope of the poset), then via detropicalization to the birational setting (here, the operation acts -- more or less -- on maps from the poset to an arbitrary field).

In the latter setting, it is no longer a priori clear even that birational rowmotion has finite order, and for many posets the order is indeed infinite. However, we show that, for the poset P = [p] × [q] (product of two chains), birational rowmotion has the same order, p+q, as ordinary rowmotion. We also show that birational (hence also ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture, of which our result can be considered an analogue.

The proofs are at most sketched in the above abstract, while the main paper offers more detail.

This talk surveys some basic results about integral-valued (a.k.a. integer-valued) polynomials (such as the one that they can be written as Z-linear combinations of binomial coefficients). It is meant for interested undergraduates.