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

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U2 - 10.12958/adm1793

DO - 10.12958/adm1793

M3 - Article

AN - SCOPUS:85139861191

VL - 33

SP - 78

EP - 91

JO - Algebra and Discrete Mathematics

JF - Algebra and Discrete Mathematics

SN - 1726-3255

IS - 2

ER -