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Diskr. Mat., 2016, Volume 28, Issue 3, Pages 59–96 (Mi dm1384)  

This article is cited in 1 scientific paper (total in 1 paper)

Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle

F. M. Malyshev

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The paper is concerned with estimating the number $\xi$ of ones in triangular arrays consisting of elements of the field $GF(2)$ which are defined by the bottom row of $s$ elements. The elements of each higher row are obtained (as in Pascal triangles) by the summation of pairs of elements from the corresponding lower row. It is shown that there exists a monotone unbounded sequence $0=k_0<k_1<k_2< ...$ of rational numbers such that, for any $k>0$, for sufficiently large $s$ the admissible values of $\xi$ which are smaller than $ks$ or larger than $s(s+1)/3-sk/3$ are concentrated in neighbourhoods of points $k_is$ and $s(s+1)/3-sk_i/3$, $i\geqslant0$. The resulting estimates of the neighbourhoods are functions of $i$ for each $i\geqslant0$ and do not depend on $s$. The distributions of the numbers of triangles with values $\xi$ in these neighbourhoods depend only on the residues of $s$ with respect to moduli that depend on $i\geqslant0$.

Keywords: Pascal triangle, (0-1)-matrix, extreme combinatorial configuration.

DOI: https://doi.org/10.4213/dm1384

Full text: PDF file (1219 kB)
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English version:
Discrete Mathematics and Applications, 2017, 27:3, 149–176

Bibliographic databases:

UDC: 519.14
Received: 17.03.2016

Citation: F. M. Malyshev, “Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle”, Diskr. Mat., 28:3 (2016), 59–96; Discrete Math. Appl., 27:3 (2017), 149–176

Citation in format AMSBIB
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\by F.~M.~Malyshev
\paper Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
\jour Diskr. Mat.
\yr 2016
\vol 28
\issue 3
\pages 59--96
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\crossref{https://doi.org/10.4213/dm1384}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3643047}
\elib{https://elibrary.ru/item.asp?id=27349816}
\transl
\jour Discrete Math. Appl.
\yr 2017
\vol 27
\issue 3
\pages 149--176
\crossref{https://doi.org/10.1515/dma-2017-0019}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85021847290}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. F. M. Malyshev, “Bulevy analogi treugolnika Paskalya s maksimalno vozmozhnym chislom edinits”, Diskret. matem., 32:1 (2020), 51–59  mathnet  crossref  mathscinet
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