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Register a new userCompositional (km,kn)-Shuffle Conjectures
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In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. In fact, they formulated one conjecture for each pair (m,n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair (km,kn), with (m,n) co-prime and k > 1.
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Consider a permutation p to be any finite list of distinct positive integers. A statistic is a function St whose domain is all permutations. Let S(p,q) be the set of shuffles of two disjoint permutations p and q. We say that St is shuffle compatible if the distribution of St over S(p,q) depends only on St(p), St(q), and the lengths of p and q. This notion is implicit in Stanleys work on P-partitions and was first explicitly studied by Gessel and Zhuang. One of the places where shuffles are useful is in describing the product in the algebra of quasisymmetric functions. Recently Adin, Gessel, Reiner, and Roichman defined an algebra of cyclic quasisymmetric functions where a cyclic version of shuffling comes into play. The purpose of this paper is to define and study cyclic shuffle compatibility. In particular, we show how one can lift shuffle compatibility results for (linear) permutations to cyclic ones. We then apply this result to cyclic descents and cyclic peaks. We also discuss the problem of finding a cyclic analogue of the major index.
Define a permutation to be any sequence of distinct positive integers. Given two permutations p and s on disjoint underlying sets, we denote by p sh s the set of shuffles of p and s (the set of all permutations obtained by interleaving the two permutations). A permutation statistic is a function St whose domain is the set of permutations such that St(p) only depends on the relative order of the elements of p. A permutation statistic is shuffle compatible if the distribution of St on p sh s depends only on St(p) and St(s) and their lengths rather than on the individual permutations themselves. This notion is implicit in the work of Stanley in his theory of P-partitions. The definition was explicitly given by Gessel and Zhuang who proved that various permutation statistics were shuffle compatible using mainly algebraic means. This work was continued by Grinberg. The purpose of the present article is to use bijective techniques to give demonstrations of shuffle compatibility. In particular, we show how a large number of permutation statistics can be shown to be shuffle compatible using a few simple bijections. Our approach also leads to a method for constructing such bijective proofs rather than having to treat each one in an ad hoc manner. Finally, we are able to prove a conjecture of Gessel and Zhuang about the shuffle compatibility of a certain statistic.
In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions $G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety of his examples could be explained by the conjecture that the Dyck path LLT polynomials exhibit the same phenomenon. In this paper we list the results of computer explorations which suggest that other examples exhibit the same phenomenon. We prove two of the resulting conjectures and propose algorithms that would prove several of our conjectures. In writing this paper we have learned that similar findings have been independently discovered by Per Alexandersson.
The Stern poset $mathcal{S}$ is a graded infinite poset naturally associated to Sterns triangle, which was defined by Stanley analogously to Pascals triangle. Let $P_n$ denote the interval of $mathcal{S}$ from the unique element of row $0$ of Sterns triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $ngeq 1$ let begin{align*} L_n(q)&=2cdotleft(sum_{k=1}^{2^n-1}A_{P_k}(q)right)+A_{P_{2^n}}(q), end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $ngeq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this paper we obtain a simple recurrence relation satisfied by $L_n(q)$ and affirmatively solve Stanleys conjectures. We also establish the asymptotic normality of the coefficients of $L_n(q)$.
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture. It is well-known that the three conjectures hold in dimensions d leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d geq 5.
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