Geodesic
Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
Diskretnaya Matematika, Tome 28 (2016) no. 3, pp. 59-96.

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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 $0=k_0$ 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. 
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F. M. Malyshev. Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle. Diskretnaya Matematika, Tome 28 (2016) no. 3, pp. 59-96. http://geodesic.mathdoc.fr/item/DM_2016_28_3_a5/

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