Maximal Lehmer Codes for Permutations with Classical and Consecutive 321-Avoidance
Abstract
Let Sn(321) and Sn(underline321) denote the sets of n-permutations avoiding the classical pattern $321$ and the consecutive pattern underline321, respectively. Permutations are in bijection with Lehmer codes, a type of inversion sequence. Using Lehmer codes, we create the corresponding weighted posets Ln(321) and Ln(underline321), where the weight of a code is the inversion number of its permutation. We show that there are 2n−2 maximal elements of Ln(321), while the maximal elements of Ln(underline321) are enumerated by the Padovan numbers. We also show that when n is even, each of these posets has a unique maximum weight element, and that when n is odd, there are two maximum weight elements. These maximum weight Lehmer codes correspond to the pattern avoiding permutations with maximum inversion number.
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BibTeX
@misc{beveridge2026maximal,
title = {{Maximal Lehmer Codes for Permutations with Classical and Consecutive 321-Avoidance}},
author = {Andrew Beveridge and Yufan Hu and Yucheng Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33137},
eprint = {2609.33137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33137},
}
RIS
TY - UNPB
TI - Maximal Lehmer Codes for Permutations with Classical and Consecutive 321-Avoidance
AU - Andrew Beveridge
AU - Yufan Hu
AU - Yucheng Liu
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33137
UR - https://arxiv.org/abs/2609.33137
AB - Let S_n(321) and S_n(underline321) denote the sets of n-permutations avoiding the classical pattern $321$ and the consecutive pattern underline321, respectively. Permutations are in bijection with Lehmer codes, a type of inversion sequence. Using Lehmer codes, we create the corresponding weighted posets L_n(321) and L_n(underline321), where the weight of a code is the inversion number of its permutation. We show that there are 2^n−2 maximal elements of L_n(321), while the maximal elements of L_n(underline321) are enumerated by the Padovan numbers. We also show that when n is even, each of these posets has a unique maximum weight element, and that when n is odd, there are two maximum weight elements. These maximum weight Lehmer codes correspond to the pattern avoiding permutations with maximum inversion number.
ER -