Statistics on bargraphs of inversion sequences of permutations

Toufik Mansour; Mark Shattuck

Statistics on bargraphs of inversion sequences of permutations

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

An inversion sequence (x₁, …, x_n) is one such that 1 ≤ x_i ≤ i for all 1 ≤ i ≤ n. We first consider the joint distribution of the area and perimeter statistics on the set I_n of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distri-bution. Explicit formulas for the generating functions are found in some special cases as are expressions for the totals of the respective statistics on I_n. A similar treatment is provided for the joint distribution on I_n for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of this latter distribution and the Stirling numbers of the first kind and Eulerian numbers.

Original language	English
Pages (from-to)	42-49
Number of pages	8
Journal	Discrete Mathematics Letters
Volume	4
State	Published - 2020

Bibliographical note

Publisher Copyright:
© 2020 the authors.

Keywords

Bargraph
Combinatorial statistic
Inversion sequence

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Cite this

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title = "Statistics on bargraphs of inversion sequences of permutations",

abstract = "An inversion sequence (x1, …, xn) is one such that 1 ≤ xi ≤ i for all 1 ≤ i ≤ n. We first consider the joint distribution of the area and perimeter statistics on the set In of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distri-bution. Explicit formulas for the generating functions are found in some special cases as are expressions for the totals of the respective statistics on In. A similar treatment is provided for the joint distribution on In for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of this latter distribution and the Stirling numbers of the first kind and Eulerian numbers.",

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T1 - Statistics on bargraphs of inversion sequences of permutations

AU - Mansour, Toufik

AU - Shattuck, Mark

PY - 2020

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N2 - An inversion sequence (x1, …, xn) is one such that 1 ≤ xi ≤ i for all 1 ≤ i ≤ n. We first consider the joint distribution of the area and perimeter statistics on the set In of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distri-bution. Explicit formulas for the generating functions are found in some special cases as are expressions for the totals of the respective statistics on In. A similar treatment is provided for the joint distribution on In for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of this latter distribution and the Stirling numbers of the first kind and Eulerian numbers.

AB - An inversion sequence (x1, …, xn) is one such that 1 ≤ xi ≤ i for all 1 ≤ i ≤ n. We first consider the joint distribution of the area and perimeter statistics on the set In of inversion sequences of length n represented as bargraphs. Functional equations for both the ordinary and exponential generating functions are derived from recurrences satisfied by this distri-bution. Explicit formulas for the generating functions are found in some special cases as are expressions for the totals of the respective statistics on In. A similar treatment is provided for the joint distribution on In for the statistics recording the number of levels, descents and ascents. Some connections are made between specific cases of this latter distribution and the Stirling numbers of the first kind and Eulerian numbers.

KW - Bargraph

KW - Combinatorial statistic

KW - Inversion sequence

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SN - 2664-2557

VL - 4

SP - 42

EP - 49

JO - Discrete Mathematics Letters

JF - Discrete Mathematics Letters

ER -

Statistics on bargraphs of inversion sequences of permutations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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