Combinatorics of positional colored compositions

TL;DR

The paper studies positional $n$-color compositions, where colors are allowed only at certain positions, using a generating-function framework to derive explicit forms for the counting sequences. It introduces and analyzes EVEN and ODD colored compositions, derives their generating functions $F_{e}(x)$ and $F_{o}(x)$, and extends the method to the general $(m,k)$-$n$-colored model with a three-case residue decomposition. A central contribution is a suite of bijections linking these colored compositions to classical objects such as ternary and boolean strings and to 321-avoiding separable permutations, including a unified $(m,k)$-color perspective. These connections yield new combinatorial proofs, identify OEIS sequences, and provide structural insight into how position-based coloring interacts with standard combinatorial families. The results deepen understanding of colored compositions and open avenues for broader generalizations and cross-domain enumerations.

Abstract

We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial objects are discussed, with combinatorial arguments provided and generalized for these observations.

Figures (4)

• Figure 1: The spotted tiling representation of $3_1$ (top) and $3_3$ (bottom).
• Figure 4: $\text{Mapping } 1+2_2+1+6_4+4 = 14 \text{ to } 00200002221111$
• Figure 5: $\text{Mapping } 1101111110000 \text{ to } 1+2_{i}+1+6_{j}+4$ where $1\leq i \leq 2 \text{ and } 1\leq j\leq6$
• Figure 6: The 321-avoiding separable permutation $(1,2,6,7,3,4,5,8,9,10,12,13,11)$ maps to $3+4_2+4+2_2$.

Theorems & Definitions (15)

• Definition 1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Corollary 1
• Theorem 3.3
• proof
• Theorem 3.4
• proof