TY - JOUR

T1 - Bijective Enumeration and Sign-Imbalance for Permutation Depth and Excedances

AU - Eu, Sen Peng

AU - Fu, Tung Shan

AU - Lo, Yuan Hsun

N1 - Publisher Copyright:
© S.-P. Eu, T.-S. Fu & Y.-H. Lo.

PY - 2024/6/24

Y1 - 2024/6/24

N2 - We present a simplified variant of Biane’s bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic so-called depth of a permutation. This generalizes a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the symmetric group Sn of permutations. In terms of weighted Motzkin path, we establish an involution on Sn that reverses the parities of depth and excedance numbers simultaneously, which proves that the numbers of permutations with even and odd depth (excedance numbers, respectively) are equal if n is even and differ by the tangent number if n is odd. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.

AB - We present a simplified variant of Biane’s bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic so-called depth of a permutation. This generalizes a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the symmetric group Sn of permutations. In terms of weighted Motzkin path, we establish an involution on Sn that reverses the parities of depth and excedance numbers simultaneously, which proves that the numbers of permutations with even and odd depth (excedance numbers, respectively) are equal if n is even and differ by the tangent number if n is odd. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.

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U2 - 10.4204/EPTCS.403.19

DO - 10.4204/EPTCS.403.19

M3 - Conference article

AN - SCOPUS:85197660811

SN - 2075-2180

VL - 403

SP - 87

EP - 91

JO - Electronic Proceedings in Theoretical Computer Science, EPTCS

JF - Electronic Proceedings in Theoretical Computer Science, EPTCS

T2 - 13th Conference on Random Generation of Combinatorial Structures. Polyominoes and Tilings, GASCom 2024

Y2 - 24 June 2024 through 28 June 2024

ER -