New bijections from $n$-color compositions

Abstract

Combinatorial bijections are given from the set of n-color compositions of ν, for which a part of size n can take on n colors, to the set of compositions of 2ν − 1 having only parts of size 1 or 2, the set of compositions of 2ν having only odd parts, and the set of compositions of 2ν + 1 having no parts of size 1. A generalized bijection based on similar ideas is then given between the set of compositions of ν into parts congruent to a(mod b) and the set of compositions of ν + b − a into parts congruent to b(mod a) with each part greater than b − a.

The combinatorics of weighted vector compositions

“…Section 5 gives divisibility properties of k ℓ f and in Section 6, we derive congruences and identities for sums of k ℓ f , including c f (ℓ). Our results in these two sections generalize corresponding results from [16,41] for weighted integer compositions, and others for ordinary binomial coefficients. We also generalize here the notion of so-called s-color compositions in which a part of size s may occur in s different colors in a composition [2].…”

“…., s N colors (for component N ). It is well-known that ordinary s-color compositions [2] are closely related to "1-2-color compositions", that is, integer compositions that only have part sizes in {1, 2}; see, e.g., Shapcott [41]. The next theorem generalizes this relationship.…”

“…, q N denote the bijective re-writings. Consider this equation in each row, s ri i + t qi i = ℓ. Encode the integer composition (s ri i , t qi i ) of ℓ into the "cross-and-dash representation" of Shapcott [41] in which crosses separate parts and a part value of size π with color 1 ≤ c ≤ π is denoted by π − 1 dashes and one cross in position c. Then, as in Shapcott [41], Proposition 2, let crosses stand for 1s and dashes for 2s. This proves the bijection between f prod -weighted compositions and g-weighted compositions.…”

S-Restricted Compositions Revisited

“…The S-restricted composition problem can also be studied as a special case of the problems weighted compositions Eger (2013Eger ( , 2015Eger ( , 2016; Whitney and Moser (1961), restricted words Janjić (2016), and colored compositions Shapcott (2013Shapcott ( , 2012. The solutions presented for these problems, however, defer the problem to finding a closed-form formula for so-called extended binomial coefficients.…”

On the Number of Many-to-Many Alignments of Multiple Sequences

“…Here, F n denotes the n-th Fibonacci number (see, e.g., [16,30]). Moreover, M ≥ 1 and φ is the unique positive real solution to 1…”