Generalized Pascal triangle for binomial coefficients of words

Abstract

Abstract. We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p.

Convergence of Pascal-like triangles in Parry–Bertrand numeration systems

“…If we normalize this compact set by a homothety of ratio 1/2 n , we get a sequence of subsets of [0, 1] × [0, 1] which converges, for the Hausdorff distance, to the Sierpiński gasket when n tends to infinity. In the extended context described above, the case when b = 2 gives similar results and the limit set, generalizing the Sierpiński gasket, is described using a simple combinatorial property called ( ) [10].…”

“…Inspired by [10], we study the sequence (S b (n)) n≥0 which counts, on each row m of P b , the number of words of L b occurring as subwords of the mth word in L b , i.e., S b (m) = #{n ∈ N | P b (m, n) > 0}. This sequence is shown to be b-regular [11,13].…”

“…In Section 2, we collect necessary background. Section 3 is devoted to a special combinatorial property that extends the ( ) condition from [10]. This new condition allows us to define a sequence of compact sets, which is shown to be a Cauchy sequence in Section 4.…”

“…Several generalizations and variations of the Pascal triangle exist and lead to interesting combinatorial, geometrical or dynamical properties [1,2,8,9,10]. This paper is inspired by a series of papers based on generalizations of Pascal triangles to finite words [10,11,12,13].…”

Computing the k-binomial Complexity of the Thue–Morse Word

“…Binomial coefficients of words have been extensively studied [19]: u x denotes the number of occurrences of x as a subword, i.e., a subsequence, of u. They have been successfully used in several applications: p-adic topology [6], non-commutative extension of Mahler's theorem on interpolation series [24], formal language theory [13], Parikh matrices, and a generalization of Sierpiński's triangle [18].…”