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The descent statistic on signed simsun permutations [article]

Shi-Mei Ma, Toufik Mansour, Hai-Na Wang
2016 arXiv   pre-print
In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents.  ...  Properties of the polynomials, including the recurrence relations and generating functions are studied.  ...  On signed simsun permutations of B n Let RB + n = {π ∈ RB n : π(1) > 0} and RB − n = {π ∈ RB n : π(1) < 0}.  ... 
arXiv:1605.02618v2 fatcat:quwm6iiebrf6vjfpoknd22awcm

The peak statistics on simsun permutations [article]

Shi-Mei Ma, Yeong-Nan Yeh
2016 arXiv   pre-print
Moreover, we introduce and study simsun permutations of the second kind.  ...  In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations.  ...  It is well known that the descent statistic is equidistributed over n-simsun permutations and n-André permutations (see [5] ), and there are bijections between simsun permutations and increasing 1-2 trees  ... 
arXiv:1601.06505v2 fatcat:wnghsutqknehfnc5yx7gqico5m

On thecd-variation polynomials of André and simsun permutations

G. Hetyei
1996 Discrete & Computational Geometry  
On the cd-Variation Polynomials of Andr6 and Simsun Permutations 261  ...  We give a new signed generalization of Andr6 permutations, together with a new notion of cd-variation for signed permutations.  ...  < j' and n(j") < ~r(j'), and the same holds when j' = n and j' -1 is a descent" On the cd-Variation Polynomials of Andr6 and Simsun Permutations  ... 
doi:10.1007/bf02711512 fatcat:ixlnfzjfwvdw3k7wtzuoiganzq

Convex Polytopes and Enumeration

Rodica Simion
1997 Advances in Applied Mathematics  
On the enumerative side, they involved ordered graphical sequences, combinatorial statistics on the symmetric and hyperoctahedral groups, lattice paths, Baxter, Andre, and simsun permutations, q-Catalan  ...  This is an expository paper on connections between enumerative combinatorics and convex polytopes.  ...  More precisely, A n, i denotes the number of signed permutations on n letters, having exactly i descents.  ... 
doi:10.1006/aama.1996.0505 fatcat:wukuqe7hkvg7hm3croxqu2enli

Mesh patterns and the expansion of permutation statistics as sums of permutation patterns [article]

Petter Brändén, Anders Claesson
2011 arXiv   pre-print
We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index.  ...  We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns.  ...  In terms of mesh patterns, a permutation is simsun if and only if it avoids the pattern simsun = . simsun permutations are central in describing the action of the symmetric group on the maximal chains  ... 
arXiv:1102.4226v2 fatcat:xgzbvobda5d4lpkgjz5geihuou

Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

Petter Brändén, Anders Claesson
2011 Electronic Journal of Combinatorics  
We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index.  ...  We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns.  ...  In terms of mesh patterns, a permutation is simsun if and only if it avoids the pattern simsun = . simsun permutations are central in describing the action of the symmetric group on the maximal chains  ... 
doi:10.37236/2001 fatcat:skswqrflhfeadmt36wp4crlvse

Springer Numbers and Arnold Families Revisited [article]

Sen-Peng Eu, Tung-Shan Fu
2021 arXiv   pre-print
Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.  ...  For the calculation of Springer numbers (of root systems) of type B_n and D_n, Arnold introduced a signed analogue of alternating permutations, called β_n-snakes, and derived recurrence relations for enumerating  ...  Let RSI n ⊂ B n be the set of signed permutations σ such that |σ| is a Simsun permutation in S n . These objects are called signed Simsun permutations of type I.  ... 
arXiv:2111.00888v1 fatcat:hf64mmesgvcdzovxlmkethcpju

Positivity and divisibility of alternating descent polynomials [article]

Zhicong Lin, Shi-Mei Ma, David G.L. Wang, Liuquan Wang
2020 arXiv   pre-print
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic.  ...  We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation.  ...  The second author was supported by the National Natural Science Foundation of China grant 12071063. The third author was supported by the National Natural Science Foundation of China grant 11671037.  ... 
arXiv:2011.02685v1 fatcat:oohhdhiecjbwhhgzz35w4vjg34

Alternating Eulerian polynomials and left peak polynomials [article]

Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh
2021 arXiv   pre-print
polynomials have gamma-vectors alternate in sign.  ...  We establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials of permutations in the symmetric group, which implies that the type B alternating Eulerian  ...  The permutation π is called simsun if for each k ∈ [n], the subword of π restricted to [k] (in the order they appear in π) contains no double descents.  ... 
arXiv:2104.09374v3 fatcat:2nochse7mffzjjr7abgza34ffi

The gamma-positivity of Eulerian polynomials and succession statistics [article]

Shi-Mei Ma, Jun Ma, Jean Yeh, Yeong-Nan Yeh
2020 arXiv   pre-print
Properties of the enumerative polynomials for permutations, signed permutations and derangements, including generating functions and gamma-positivity are studied, which generalize and unify earlier results  ...  This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics.  ...  In the next section, we first count permutations in S n by the numbers of big ascents, descents and successions. We then consider the joint distribution of sextuple statistics on B n .  ... 
arXiv:2002.06930v2 fatcat:h3v6gpi54bg4bb3xjzf66zckfq

Variations on Descents and Inversions in Permutations [article]

Denis Chebikin
2008 arXiv   pre-print
We study new statistics on permutations that are variations on the descent and the inversion statistics.  ...  We show that this statistic is equidistributed with the 3-descent set statistic on permutations sigma = sigma_1sigma_2...sigma_n+1 with sigma_1 = 1, defined to be the set of indices i such that the triple  ...  Acknowledgments This paper is part of the author's Ph.D. thesis. I would like to thank Pavlo Pylyavskyy for his ideas and conversations that led to this work.  ... 
arXiv:0804.1935v1 fatcat:u3uiim7k4bdq5cnkduxlkso2qu

Signed Euler-Mahonian identities [article]

Sen-Peng Eu, Zhicong Lin, Yuan-Hsun Lo
2020 arXiv   pre-print
Some obtained identities can be further restricted on some particular set of permutations. We also derive some new interesting sign-balance polynomials for types B_n and D_n.  ...  By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types B_n, D_n, and the complex reflection group G(r,1,n), where the 'sign' is taken to be any one-dimensional  ...  Acknowledgments The authors would like to express their gratitude to the referees for their valuable comments and suggestions on improving the presentation of this paper.  ... 
arXiv:2007.13176v1 fatcat:ivui4ddvubf63hhnhc3orlf3wq

The peak statistics on simsun permutations

Shi-Mei Ma, Yeong-Nan Yeh
unpublished
Moreover, we introduce and study simsun permutations of the second kind.  ...  In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations.  ...  Acknowledgements The authors thank the referees for their valuable suggestions which lead to a substantial improvement of the paper.  ... 
fatcat:mhy5tuh7wbflhf5rsuvaso775i

Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations [article]

Megan A. Martinez, Carla D. Savage
2018 arXiv   pre-print
We highlight open questions about the relationship between pattern avoiding inversion sequences and families such as plane permutations and Baxter permutations.  ...  We show that "avoiding a triple of relations" can characterize inversion sequences with a variety of monotonicity or unimodality conditions, or with multiplicity constraints on the elements.  ...  We once again extend sincere thanks to Neil Sloane and the OEIS Foundation, Inc. The On-Line Encyclopedia of Integer Sequences [15] was an invaluable resource for this project.  ... 
arXiv:1609.08106v2 fatcat:ldrdsumewzabrp356izymtt274

Enumeration of snakes and cycle-alternating permutations [article]

Matthieu Josuat-Vergès
2010 arXiv   pre-print
Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations.  ...  We obtain the generating functions, in terms of trigonometric functions for exponential ones and in terms of J-fractions for ordinary ones.  ...  Acknowledgement Part of this research was done during a visit of the LABRI in Bordeaux, and I thank all the Bordelais for welcoming me and for various suggestions concerning this work.  ... 
arXiv:1011.0929v1 fatcat:h47wf7idebd2ng5cihsq7f2t6m
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