957 research outputs found

    Descent polynomials for permutations with bounded drop size

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    Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,...,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.Comment: 15 page

    Carries, shuffling, and symmetric functions

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    The "carries" when n random numbers are added base b form a Markov chain with an "amazing" transition matrix determined by Holte. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.Comment: 23 page

    Recurrences for Eulerian polynomials of type B and type D

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    We introduce new recurrences for the type B and type D Eulerian polynomials, and interpret them combinatorially. These recurrences are analogous to a well-known recurrence for the type A Eulerian polynomials. We also discuss their relationship to polynomials introduced by Savage and Visontai in connection to the real-rootedness of the corresponding Eulerian polynomials

    An affine generalization of evacuation

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    We establish the existence of an involution on tabloids that is analogous to Schutzenberger's evacuation map on standard Young tableaux. We find that the number of its fixed points is given by evaluating a certain Green's polynomial at q=−1q = -1, and satisfies a "domino-like" recurrence relation.Comment: 32 pages, 7 figure

    Revstack sort, zigzag patterns, descent polynomials of tt-revstack sortable permutations, and Steingr\'imsson's sorting conjecture

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    In this paper we examine the sorting operator T(LnR)=T(R)T(L)nT(LnR)=T(R)T(L)n. Applying this operator to a permutation is equivalent to passing the permutation reversed through a stack. We prove theorems that characterise tt-revstack sortability in terms of patterns in a permutation that we call zigzagzigzag patterns. Using these theorems we characterise those permutations of length nn which are sorted by tt applications of TT for t=0,1,2,n−3,n−2,n−1t=0,1,2,n-3,n-2,n-1. We derive expressions for the descent polynomials of these six classes of permutations and use this information to prove Steingr\'imsson's sorting conjecture for those six values of tt. Symmetry and unimodality of the descent polynomials for general tt-revstack sortable permutations is also proven and three conjectures are given
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