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

Abstract

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 < k 1 < k 2 < ... of rational numbers such that, for any k > 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 kis and s(s + 1)/3 − ski /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.