Baxter Structures and the Odd-Even Enumeration of Red-Blue Motzkin Paths
Abstract
We study Motzkin paths satisfying the red-blue rule introduced by Wu, Eu, Ku and Shih. We answer their Question 8.5 and exhibit two new Baxter structures. Let ℳm denote the set of such paths of length m and am=|ℳm|. We give a combinatorial proof of a2n−1=A005802(n), a2n=A216947(n) (n≥ 1). For the odd subsequence, we construct a bijection with $1234$-avoiding permutations which sends the initial up and terminal down runs to the initial decreasing runs of a permutation and its inverse. The boundary correspondence arising from this bijection further identifies red-blue Motzkin paths of length 2n with $1234$-avoiding permutations of length n+1 in which $1$ occurs to the left of $2$. We establish parallel binomial decompositions of this restricted permutation class and of $2$-colored noncrossing set partitions. The two decompositions have the same binomial coefficients, and their respective cores are explicitly in bijection with non-intersecting triples of lattice paths, a classical Baxter family. This establishes the even identity. Moreover, we construct a second bijection between odd-length red-blue Motzkin paths and $1234$-avoiding permutations. Under this bijection, splitting a path at its ground-level blue horizontal steps corresponds to the skew decomposition of the associated permutation.
Related
- Richardson tableaux and Motzkin paths
- Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity
- Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra
- Restricted (Relative) Rota-Baxter operators on restricted Lie algebras and restricted Lie triple systems and related structures
- Fractal Hyper-Trees: Combinatorial Enumeration, Symmetry Properties, and Ultrametric Structures
BibTeX
@misc{chen2026baxter,
title = {{Baxter Structures and the Odd-Even Enumeration of Red-Blue Motzkin Paths}},
author = {Joanna N. Chen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06599},
eprint = {2611.06599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06599},
}
RIS
TY - UNPB
TI - Baxter Structures and the Odd-Even Enumeration of Red-Blue Motzkin Paths
AU - Joanna N. Chen
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06599
UR - https://arxiv.org/abs/2611.06599
AB - We study Motzkin paths satisfying the red-blue rule introduced by Wu, Eu, Ku and Shih. We answer their Question 8.5 and exhibit two new Baxter structures. Let ℳ_m denote the set of such paths of length m and a_m=|ℳ_m|. We give a combinatorial proof of a_2n−1=A005802(n), a_2n=A216947(n) (n≥ 1). For the odd subsequence, we construct a bijection with $1234$-avoiding permutations which sends the initial up and terminal down runs to the initial decreasing runs of a permutation and its inverse. The boundary correspondence arising from this bijection further identifies red-blue Motzkin paths of length 2n with $1234$-avoiding permutations of length n+1 in which $1$ occurs to the left of $2$. We establish parallel binomial decompositions of this restricted permutation class and of $2$-colored noncrossing set partitions. The two decompositions have the same binomial coefficients, and their respective cores are explicitly in bijection with non-intersecting triples of lattice paths, a classical Baxter family. This establishes the even identity. Moreover, we construct a second bijection between odd-length red-blue Motzkin paths and $1234$-avoiding permutations. Under this bijection, splitting a path at its ground-level blue horizontal steps corresponds to the skew decomposition of the associated permutation.
ER -