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Sibirskii Matematicheskii Zhurnal, 2022, Volume 63, Number 2, Pages 316–333
DOI: https://doi.org/10.33048/smzh.2022.63.205 (Mi smj7659) 		 
This article is cited in 1 scientific paper (total in 1 paper)

Values of the permanent function on multidimensional $(0,1)$-matrices

A. E. Gutermanabc, I. M. Evseevab, A. A. Taranenkod

a Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics
c Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region
d Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
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DOI: https://doi.org/10.33048/smzh.2022.63.205
Abstract: We study the range of the permanent function for the multidimensional matrices of $0$ and $1$. The main result is a multidimensional version for the Brualdi–Newman upper bound on the consecutive values of the permanent (1965). Moreover, we deduce a formula for the permanent of the multidimensional $(0,1)$-matrices through the number of partial zero diagonals. Using the formula, we evaluate the permanents of the $(0,1)$-matrices with a few zeros and estimate the permanents of the matrices whose all zero entries are located in several orthogonal hyperplanes. We consider some divisibility properties of the permanent and illustrate the results by studying the values of the permanent for the $3$-dimensional $(0,1)$-matrices of order $3$.
Keywords: permanent, multidimensional matrix, $(0,1)$-matrix, Brualdi–Newman theorem.
Funding agency Grant number
Siberian Branch of Russian Academy of Sciences I.5.1, проект № 0314-2019-0016
A. A. Taranenko (Theorem 4.8 and Proposition 7.4) was partially supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant no. I.5.1, Project 0314–2019–0016).
Received: 31.05.2021
Revised: 21.09.2021
Accepted: 11.10.2021
English version:
Siberian Mathematical Journal, 2022, Volume 63, Issue 2, Pages 262–276
DOI: https://doi.org/10.1134/S0037446622020057
Document Type: Article
UDC: 512.643+519.142
MSC: 35R30
Language: Russian
Citation: A. E. Guterman, I. M. Evseev, A. A. Taranenko, “Values of the permanent function on multidimensional $(0,1)$-matrices”, Sibirsk. Mat. Zh., 63:2 (2022), 316–333; Siberian Math. J., 63:2 (2022), 262–276
Citation in format AMSBIB
\Bibitem{GutEvsTar22}
\by A.~E.~Guterman, I.~M.~Evseev, A.~A.~Taranenko
\paper Values of the permanent function on multidimensional $(0,1)$-matrices
\jour Sibirsk. Mat. Zh.
\yr 2022
\vol 63
\issue 2
\pages 316--333
\mathnet{http://mi.mathnet.ru/smj7659}
\crossref{https://doi.org/10.33048/smzh.2022.63.205}
\transl
\jour Siberian Math. J.
\yr 2022
\vol 63
\issue 2
\pages 262--276
\crossref{https://doi.org/10.1134/S0037446622020057}
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