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

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

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English version:
Discrete Mathematics and Applications, 2017, 27:3, 149–176

Bibliographic databases:

Document Type: Article
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

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