Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
Фeдор Михайлович Малышев · Discrete Mathematics and Applications · 2017
Abstract The paper is concerned with estimating the number ξ 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 0, for sufficiently large s the admissible values of ξ which are smaller than ks or larger than s ( s + 1)/3 − sk /3 are concentrated in neighbourhoods of points k i s and s ( s + 1)/3 − sk i /3, i ⩾ 0. The resulting estimates of the neighbourhoods are functions of i for each i ⩾ 0 and do not depend on s . The distributions of the numbers of triangles with values ξ in these neighbourhoods depend only on the residues of s with respect to moduli that depend on i ⩾ 0.