We give short elementary expositions of combinatorial proofs of some variants of Euler's partition identity that were first addressed analytically by George Andrews, and later combinatorially by ...others. The method using certain matrices to concisely explain these bijections, based on ideas first used in a previous manuscript by the author, enables us to also give new generalizations of two of these results.
A simple model of trees for unicellular maps Chapuy, Guillaume; Féray, Valentin; Fusy, Éric
Journal of combinatorial theory. Series A,
November 2013, 2013-11-00, 2013-11-01, Letnik:
120, Številka:
8
Journal Article
Recenzirano
Odprti dostop
We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of ...Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the “recursive part” of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure.
All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer–Zagier recurrence formula, the Lehman–Walsh formula and the Goupil–Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-3-constellations.
Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories, and recently Iyama, Nakaoka and Palu ...investigated Auslander-Reiten theory in terms of Auslander-Reiten-Serre duality in extriangulated categories. In this paper, we introduce the notion of Serre duality, as a special type of Auslander-Reiten-Serre duality, and then give an equivalent condition for the existence of Serre duality. On the other hand, we study a bijection triangle in extriangulated categories which involves the restricted Auslander bijection and Auslander-Reiten-Serre duality. Let R be a commutative artinian ring. We show that the restricted Auslander bijection holds true in Hom-finite R-linear Krull-Schmidt extriangulated categories having Auslander-Reiten-Serre duality, and especially obtain the Auslander bijection in Hom-finite R-linear Krull-Schmidt extriangulated categories having Serre duality. We also give some applications, and in particular, we show that a conjecture given by Ringel holds true in this setting.
In 2015, Remmel and Wilson proved the following conjecture of Jim Haglund for permutations∑π∈Snqinv(π)∏j∈Des(π)(1+zq1+inv□,j(π))=∑π∈Snqmaj(π)∏j=1des(π)(1+zqj). Wilson extended this identity to words ...in 2016. In this note, we prove that this identity holds for set partitions.
The automatic projection filter is a recently developed numerical method for projection filtering that leverages sparse-grid integration and automatic differentiation. However, its accuracy is highly ...sensitive to the accuracy of the cumulant-generating function computed via the sparse-grid integration, which in turn is also sensitive to the choice of the bijection from the canonical hypercube to the state space. In this article, we propose two new adaptive parametric bijections for the automatic projection filter. The first bijection relies on the minimization of Kullback-Leibler divergence, whereas the second method employs the sparse-grid Gauss-Hermite quadrature. The two new bijections allow the sparse-grid nodes to adaptively move within the high-density region of the state space, resulting in a substantially improved approximation while using only a small number of quadrature nodes. The practical applicability of the methodology is illustrated in three simulated nonlinear filtering problems.
A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue ...of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuang's generalizations (Zhuang 2017 29) using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng (2012) 21 and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the γ-coefficients of the inversion polynomials restricted on 321-avoiding permutations.
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