Further combinatorial results for the symmetric inverse monoid∗

Abdallah Laradji; Abdullahi Umar

doi:10.12958/adm1793

Further combinatorial results for the symmetric inverse monoid^∗

Abdallah Laradji
, Abdullahi Umar

Mathematics

King Fahd University for Petroleum and Minerals (KFUPM)

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

Abstract

Let I_n be the set of partial one-to-one transformations on the chain X_n = {1, 2, …, n} and, for each α in I_n, let h(α) = |Imα|, f(α) = |{x ∈ X_n: xα = x}| and w(α) = max(Imα). In this note, we obtain formulae involving binomial coeffcients of F(n; p, m, k) = |{α ∈ I_n: h(α) = p ∧f(α) = m ∧ w(α) = k}| and F(n; ·, m, k) = |{α ∈ I_n: f(α) = m ∧w(α) = k}| and analogous results on the set of partial derangements of I_n .

Original language	British English
Pages (from-to)	78-91
Number of pages	14
Journal	Algebra and Discrete Mathematics
Volume	33
Issue number	2
DOIs	https://doi.org/10.12958/adm1793
State	Published - 2022

Keywords

(left) waist of α
(partial) derangement
fix of α
height of α
partial one-to-one transformation
permutation
symmetric inverse monoid

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10.12958/adm1793

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@article{d2d1923e9e244578ae894a07f2f95de5,

title = "Further combinatorial results for the symmetric inverse monoid∗",

abstract = "Let In be the set of partial one-to-one transformations on the chain Xn = \{1, 2, …, n\} and, for each α in In, let h(α) = |Imα|, f(α) = |\{x ∈ Xn: xα = x\}| and w(α) = max(Imα). In this note, we obtain formulae involving binomial coeffcients of F(n; p, m, k) = |\{α ∈ In: h(α) = p ∧f(α) = m ∧ w(α) = k\}| and F(n; ·, m, k) = |\{α ∈ In: f(α) = m ∧w(α) = k\}| and analogous results on the set of partial derangements of In .",

keywords = "(left) waist of α, (partial) derangement, fix of α, height of α, partial one-to-one transformation, permutation, symmetric inverse monoid",

author = "Abdallah Laradji and Abdullahi Umar",

note = "Funding Information: ∗The authors would like to acknowledge support from King Fahd University of Petroleum \& Minerals and Khalifa University of Science and Technology, respectively. 2020 MSC: 20M18, 20M20, 05A10, 05A15. Key words and phrases: partial one-to-one transformation, symmetric inverse monoid, height of α, fix of α, (left) waist of α, permutation, (partial) derangement. Publisher Copyright: {\textcopyright} Algebra and Discrete Mathematics.",

year = "2022",

doi = "10.12958/adm1793",

language = "British English",

volume = "33",

pages = "78--91",

journal = "Algebra and Discrete Mathematics",

issn = "1726-3255",

publisher = "Lugansk Taras Shevchenko National University",

number = "2",

}

TY - JOUR

T1 - Further combinatorial results for the symmetric inverse monoid∗

AU - Laradji, Abdallah

AU - Umar, Abdullahi

N1 - Funding Information: ∗The authors would like to acknowledge support from King Fahd University of Petroleum & Minerals and Khalifa University of Science and Technology, respectively. 2020 MSC: 20M18, 20M20, 05A10, 05A15. Key words and phrases: partial one-to-one transformation, symmetric inverse monoid, height of α, fix of α, (left) waist of α, permutation, (partial) derangement. Publisher Copyright: © Algebra and Discrete Mathematics.

PY - 2022

Y1 - 2022

N2 - Let In be the set of partial one-to-one transformations on the chain Xn = {1, 2, …, n} and, for each α in In, let h(α) = |Imα|, f(α) = |{x ∈ Xn: xα = x}| and w(α) = max(Imα). In this note, we obtain formulae involving binomial coeffcients of F(n; p, m, k) = |{α ∈ In: h(α) = p ∧f(α) = m ∧ w(α) = k}| and F(n; ·, m, k) = |{α ∈ In: f(α) = m ∧w(α) = k}| and analogous results on the set of partial derangements of In .

AB - Let In be the set of partial one-to-one transformations on the chain Xn = {1, 2, …, n} and, for each α in In, let h(α) = |Imα|, f(α) = |{x ∈ Xn: xα = x}| and w(α) = max(Imα). In this note, we obtain formulae involving binomial coeffcients of F(n; p, m, k) = |{α ∈ In: h(α) = p ∧f(α) = m ∧ w(α) = k}| and F(n; ·, m, k) = |{α ∈ In: f(α) = m ∧w(α) = k}| and analogous results on the set of partial derangements of In .

KW - (left) waist of α

KW - (partial) derangement

KW - fix of α

KW - height of α

KW - partial one-to-one transformation

KW - permutation

KW - symmetric inverse monoid

UR - https://www.scopus.com/pages/publications/85139861191

U2 - 10.12958/adm1793

DO - 10.12958/adm1793

M3 - Article

AN - SCOPUS:85139861191

SN - 1726-3255

VL - 33

SP - 78

EP - 91

JO - Algebra and Discrete Mathematics

JF - Algebra and Discrete Mathematics

IS - 2

ER -

Further combinatorial results for the symmetric inverse monoid^∗

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Keywords

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