Enumeration and asymptotics of restricted compositions having the same number of parts

Abstract

International audienceWe study pairs and m-tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of parts. Under the uniform probability model, we obtain the asymptotics for the probability that two or, more generally, m randomly and independently chosen compositions of n have the same number of parts. For a large class of compositions, we show how a nice interplay between complex analysis and probability theory allows to get full asymptotics for this probability. Our results extend an earlier work of Bóna and Knopfmacher. While we restrict our attention to compositions, our approach is also of interest for tuples of other combinatorial structures having the same number of parts

Analytic combinatorics in several variables

“…Compute the asymptotics of the coefficents on the leading diagonal. Compare the results and methods with those in Banderier and Hitczenko (2012). 13.4 (Safonov's vanishing numerator) It is expected (as explained in Section 13.2 that if we lift an algebraic function A to a rational one R, the numerator of R must vanish at a contributing critical point.…”

On the number of simultaneous core partitions with d-distinct parts

“…This proof can be thought of as exhibiting a bijection between X d,1,s and the set of compositions of s + d − 1 into parts of sizes 1 and d + 1. Formally, given a subset A ⊆ Z + , an A-restricted composition of a nonegative integer n is a finite sequence of elements of A that sum to n. These compositions have been studied in a variety of settings (see, e.g., [12], [17], [9]), and Chinn and Heubach [11] have paid special attention to the case A = {1, k}. All of our results for N d,1 (s) apply equally well to the number of {1, d + 1}-restricted compositions of s + d − 1.…”

Bandwidth efficient multi-level MPPM encoding decoding algorithms for joint brightness-rate control in VLC systems

“…The enumeration of number of symbols for given values of parameters n, l max and w is equivalent to evaluating the restricted compositions of integer w with n parts and maximum value of each part upper bounded by l max . Most of the results on restricted compositions reported in literature require all the n parts to be strictly positive integers (see [10], [11] and references there in). On the other hand the proposed solution allows an arbitrary number of zeros to be included as parts of an n part composition, which are termed as weak restricted compositions.…”