Compositions with restricted parts

Compositions With m Distinct Parts

Compositions of Integers with Bounded Parts

In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function describing the number of k-tuples whose entries are bounded in this way and sum to a fixed value g.

S-Restricted Compositions Revisited

An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in recent years. This paper is an attempt at finding a closed-form formula for the number of S-restricted compositions of n. To do so, we reduce the problem to...

Locally Restricted Compositions I. Restricted Adjacent Differences

We study compositions of the integer n in which the first part, successive differences, and the last part are constrained to lie in prescribed sets L,D,R, respectively. A simple condition on D guarantees that the generating function f(x,L,D,R) has only a simple pole on its circle of convergence and this at r(D), a function independent of L and R. Thus the number of compositions is asymptotic to...

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based...