Counting permutations by the number of vertical edges in their bargraphs

Aubrey Blecher; Charlotte Brennan; Arnold Knopfmacher; Toufik Mansour; Mark Shattuck

doi:10.25596/jalc-2020-279

Counting permutations by the number of vertical edges in their bargraphs

Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, Toufik Mansour, Mark Shattuck

Department of Mathematics

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

Abstract

In this paper, we consider a new statistic on the set S_n of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having S_n as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of S_n equals the number of runs in the members of S_n+1 by constructing an explicit bijection which applies more generally to multi-permutations.

Original language	English
Pages (from-to)	279-290
Number of pages	12
Journal	Journal of Automata, Languages and Combinatorics
Volume	25
Issue number	4
DOIs	https://doi.org/10.25596/jalc-2020-279
State	Published - 2020

Bibliographical note

Publisher Copyright:
© Institut für Informatik · Justus-Liebig-Universität Giessen.

Keywords

Bargraphs
Generating functions
Multi-set permutations
Permutation statistics

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Access to Document

10.25596/jalc-2020-279

Cite this

@article{b81e856e035645b5bb6e5c4154bca9db,

title = "Counting permutations by the number of vertical edges in their bargraphs",

abstract = "In this paper, we consider a new statistic on the set Sn of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having Sn as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of Sn equals the number of runs in the members of Sn+1 by constructing an explicit bijection which applies more generally to multi-permutations.",

keywords = "Bargraphs, Generating functions, Multi-set permutations, Permutation statistics",

author = "Aubrey Blecher and Charlotte Brennan and Arnold Knopfmacher and Toufik Mansour and Mark Shattuck",

note = "Publisher Copyright: {\textcopyright} Institut f{\"u}r Informatik · Justus-Liebig-Universit{\"a}t Giessen.",

year = "2020",

doi = "10.25596/jalc-2020-279",

language = "English",

volume = "25",

pages = "279--290",

journal = "Journal of Automata, Languages and Combinatorics",

issn = "1430-189X",

publisher = "Institut fur Informatik, Justus-Liebig-Universitat Giessen",

number = "4",

}

TY - JOUR

T1 - Counting permutations by the number of vertical edges in their bargraphs

AU - Blecher, Aubrey

AU - Brennan, Charlotte

AU - Knopfmacher, Arnold

AU - Mansour, Toufik

AU - Shattuck, Mark

N1 - Publisher Copyright: © Institut für Informatik · Justus-Liebig-Universität Giessen.

PY - 2020

Y1 - 2020

N2 - In this paper, we consider a new statistic on the set Sn of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having Sn as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of Sn equals the number of runs in the members of Sn+1 by constructing an explicit bijection which applies more generally to multi-permutations.

AB - In this paper, we consider a new statistic on the set Sn of permutations of length n which records the number of vertical edges within their bargraph representations. In-deed, we determine the distribution of this statistic on a class of multi-permutations having Sn as a subset. We compute an explicit formula for the total number of vertical edges within all members of this class and also for the number of interior vertices in the corresponding bargraphs. A generating function formula is determined and some closely related statistics are considered. Finally, we show that the total number of vertical edges in all the members of Sn equals the number of runs in the members of Sn+1 by constructing an explicit bijection which applies more generally to multi-permutations.

KW - Bargraphs

KW - Generating functions

KW - Multi-set permutations

KW - Permutation statistics

UR - http://www.scopus.com/inward/record.url?scp=85100103286&partnerID=8YFLogxK

U2 - 10.25596/jalc-2020-279

DO - 10.25596/jalc-2020-279

M3 - Article

AN - SCOPUS:85100103286

SN - 1430-189X

VL - 25

SP - 279

EP - 290

JO - Journal of Automata, Languages and Combinatorics

JF - Journal of Automata, Languages and Combinatorics

IS - 4

ER -

Counting permutations by the number of vertical edges in their bargraphs

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this