@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 = "Institute of Applied Mathematics And Mechanics of the National Academy of Sciences of Ukraine",
number = "2",
}