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A uniform double-factorial construction for consecutive-pattern packings
Abstract
For every overlap m≥ 3 and every pattern length K≥ 2m+1, we construct an explicit permutation p∈SK such that the number of permutations containing j prescribed occurrences of p, with successive starts separated by K−m, is (2j−1)!! for every j≥ 1. Additional occurrences are permitted. We prove that the corresponding occurrence poset is the ordinal sum of a chain on m+j(K−m−2) elements and a comb on 2j elements. An insertion bijection counts the linear extensions of the comb. This gives the uniform representatives requested in the double-factorial part of Question 6 in Section 7 of Blitvić, Kammoun and Steingrímsson, arXiv:2610.29846v1. A Lean 4 formalization and its recorded verification data accompany the paper.
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BibTeX
@misc{nagy2026uniform,
title = {{A uniform double-factorial construction for consecutive-pattern packings}},
author = {Zoltán Nagy},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06937},
eprint = {2611.06937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06937},
}
RIS
TY - UNPB
TI - A uniform double-factorial construction for consecutive-pattern packings
AU - Zoltán Nagy
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.06937
UR - https://arxiv.org/abs/2611.06937
AB - For every overlap m≥ 3 and every pattern length K≥ 2m+1, we construct an explicit permutation p∈S_K such that the number of permutations containing j prescribed occurrences of p, with successive starts separated by K−m, is (2j−1)!! for every j≥ 1. Additional occurrences are permitted. We prove that the corresponding occurrence poset is the ordinal sum of a chain on m+j(K−m−2) elements and a comb on 2j elements. An insertion bijection counts the linear extensions of the comb. This gives the uniform representatives requested in the double-factorial part of Question 6 in Section 7 of Blitvić, Kammoun and Steingrímsson, arXiv:2610.29846v1. A Lean 4 formalization and its recorded verification data accompany the paper.
ER -