On the Tutte polynomial of series-parallel posets
Abstract
The Tutte polynomial is a bivariate polynomial that has been extensively studied in graph and matroid theory. Gordon was the first to study the Tutte polynomial T(P;x,y) of the greedoid induced by a poset P, including the special case of series-parallel posets. In particular, Gordon and McMahon conjectured that, for any two series-parallel posets P and Q, the equality T(P;x,y)=T(Q;x,y) holds if and only if P≅ Q. In studying this conjecture, Gordon introduced a subclass 𝒫 of series-parallel posets and proved that this equivalence holds for all P,Q∈𝒫. In this paper, we introduce a new subclass H of series-parallel posets and prove that 𝒫⊆H. Moreover, we show that, for every P∈H and every Q∈SP, the equality T(P;x,y)=T(Q;x,y) holds if and only if P≅ Q, thereby extending Gordon's result.
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BibTeX
@misc{luo2026tutte,
title = {{On the Tutte polynomial of series-parallel posets}},
author = {Jianxuan Luo and Tingzeng Wu and Hong-Jian Lai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27524},
eprint = {2609.27524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27524},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Tutte polynomial of series-parallel posets
AU - Jianxuan Luo
AU - Tingzeng Wu
AU - Hong-Jian Lai
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27524
UR - https://arxiv.org/abs/2609.27524
AB - The Tutte polynomial is a bivariate polynomial that has been extensively studied in graph and matroid theory. Gordon was the first to study the Tutte polynomial T(P;x,y) of the greedoid induced by a poset P, including the special case of series-parallel posets. In particular, Gordon and McMahon conjectured that, for any two series-parallel posets P and Q, the equality T(P;x,y)=T(Q;x,y) holds if and only if P≅ Q. In studying this conjecture, Gordon introduced a subclass 𝒫 of series-parallel posets and proved that this equivalence holds for all P,Q∈𝒫. In this paper, we introduce a new subclass H of series-parallel posets and prove that 𝒫⊆H. Moreover, we show that, for every P∈H and every Q∈SP, the equality T(P;x,y)=T(Q;x,y) holds if and only if P≅ Q, thereby extending Gordon's result.
ER -