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We present several classes of information about a tree that can be recovered from its generalized degree polynomial (as defined by Crew) and hence its chromatic symmetric function, such as the number of edges connecting vertices of specified degrees, and the number of vertices of a specified degree adjacent to a specified number of leaves.
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In this appendix, we use Kronecker coefficients to resolve a Schur positivity conjecture related to a map on quasisymmetric functions arising from a generalization of the ab-index.
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We exhibit a birational map between labelings of a rectangular and trapezoidal poset that is equivariant with respect to birational rowmotion, proving that birational rowmotion on the trapezoid has finite order. We also use this map to give a new bijective proof that these posets have the same number of plane partitions of fixed height, as shown by Proctor.
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We prove a universal property for the shuffle algebra of compositions in the category of graded connected Hopf algebras equipped with an infinitesimal character. We then give general constructions for quasisymmetric power sums, recovering four previous constructions from the literature, and study their properties.
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We use toggles to construct piecewise-linear and birational versions of Rubey's bijections between fillings of moon polyominoes that preserve certain chain statistics. These results imply Ehrhart equivalence and Ehrhart quasi-polynomial period collapse of certain rational polytopes associated to moon polyominoes.
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We show how a formula for iterated birational rowmotion on a rectangular poset can be derived by way of the octahedron recurrence, the Dodgson condensation formula, and the Lindström-Gessel-Viennot Lemma. We also discuss connections to birational RSK and Greene's Theorem.
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We show that twisted Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i + \partial_i) \widetilde{\mathfrak S}_w$ are monomial positive. We also show how skew divided difference operators can be used to prove the Pieri rule for Schubert polynomials.
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We show how to construct a polytope whose integer point transform projects to the Schubert polynomial $\mathfrak S_w$ if $w$ has a column-convex Rothe diagram.
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We show that for a permutation $w$ in a certain pattern avoidance class, the Schubert polynomial $\mathfrak S_w$ has a determinantal formula in terms of elementary symmetric polynomials.
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We show that the algebra of Schur operators $u_i$ and $d_i$ on Young's lattice is defined by quadratic relations.
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We prove that the $P$-partition generating function of a connected, naturally labeled poset is irreducible. We also give a rule for the expansion of a $P$-partition generating function into type 1 quasisymmetric power sums that generalizes the Murnaghan-Nakayama rule.
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We investigate the power of $2$ that divides the number of perfect matchings of a graph. In particular, we give a fast algorithm for determining whether the number of domino tilings of a simply connected planar region is even or odd based on billiard paths.
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We give a complete list of relations between Schur operators $u_i$, which act on partitions by adding a box to column $i$ (if possible).
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We discuss conditions under which two naturally labeled posets can have the same partition generating function.
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We show that Gelfand-Tsetlin polytopes can be expressed as flow polytopes. We also generalize this result to certain marked order polytopes of strongly planar posets.
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We consider a generalization of Haglund's formula for the Hilbert series of the space of diagonal harmonics as a sum over Tesler matrices by summing over flow polytopes of threshold graphs.
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We show how the theory of noncommutative super Schur functions (based on work of Fomin and Greene) gives a combinatorial formula for the Kronecker coefficient $g_{\lambda\mu\nu}$ when $\mu$ is a hook.
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We give a simple combinatorial rule for the Kronecker coefficient $g_{\lambda\mu\nu}$ when $\mu$ is a hook.
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We define subalgebras of the Fomin-Kirillov algebra for any finite graph and show that they have a surprising number of nice properties analogous to those of Coxeter groups and nil-Coxeter algebras. We explicitly compute the Hilbert series for the subalgebras corresponding to the Dynkin diagrams $A_n$, $D_n$, $E_n$ ($n=6,7,8$), and $\tilde A_{n-1}$.
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We prove that the commutative quotient of the Fomin-Kirillov algebra of a graph $G$ on $n$ vertices is isomorphic to the Orlik-Terao algebra of $G$. In particular, its Hilbert series is $(-t)^n \cdot \chi_G(-t^{-1})$, where $\chi_G(t)$ is the chromatic polynomial of $G$.
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We give a combinatorial criterion for when the Specht module of an arbitrary diagram admits a (complete) branching rule. We also show that the only relations in such Specht modules are given by generalized Garnir relations.
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We give a positive formula for the skew divided difference operators (defined by Macdonald) in terms of divided difference operators $\partial_{ij}$ with $i < j$, settling a conjecture of Kirillov.
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We give an explicit description of the coefficients of $(1-x)\Phi_{pqr}(x)$. We also define an analogous polynomial for any number of primes and describe their coefficients and growth rate as the number of primes increases.
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We give a simple proof of a result of Erman, Smith, and Várilly-Alvarado relating Laurent polynomials and Eulerian numbers using Bernstein's theorem. We also show that a refinement of the Eulerian numbers gives a combinatorial interpretation for volumes of certain (rational) hyperplane sections of the hypercube.
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We show that the set of hypergraph degree sequences is not the intersection of a lattice and a convex polytope. We also prove an analogous result for multipartite hypergraphs.
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We prove that the normalized volume of the matching polytope of a forest equals the dimension of the corresponding Specht module. We also give $S_n$- and $GL_n$-branching rules, thereby defining a notion of Schur functions and standard/semistandard tableaux for forests.
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We study the number of matrices of a given rank with specified zero entries over a finite field. We show that these numbers give a $q$-analogue of certain restricted permutations and obtain explicit $q$-analogues for derangement numbers. We also consider the analogous questions for symmetric and skew-symmetric cases and discuss related questions.
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We present a conjecture relating the cohomology class of certain subvarieties of the Grassmannian to the structure of certain representations of the symmetric group and give evidence towards this conjecture.
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We give a Littlewood-Richardson rule based on iteratively deforming a skew Young diagram into a straight shape. This rule is based on a geometric rule by Izzet Coskun.
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We consider the problem of counting subrings of $\mathbf Z^n$ of a given index $k$. We show that a decomposition theorem holds and give a precise result when $n$ is at most 4 or $k$ is not divisible by the sixth power of any prime.
