Segre powers of posets preserve EL-shellability

Abstract

For a bounded and graded poset $P$, we show that if $P$ is EL-shellable, then so is its $t$-fold Segre power $P^{(t)}=P\circ \cdots \circ P$ ($t$ factors), as defined by Björner and Welker [J. Pure Appl. Algebra, 198(1-3), 43--55 (2005)]. Our EL-labeling leads to formulas for the rank-selected invariants of $P^{(t)}$, generalising those given by Stanley for the subspace lattice [J. Combinatorial Theory Ser. A, 20(3):336-356, 1976].

9 pages, 3 figures. This paper contains the results of Section 2, Theorems 2.6 and 2.7 of arXiv2408.08421v2 (Version 2), which was was split into two parts. The main part (25 pages) has already appeared in the journal Enumer. Comb. Appl. 5 (2025), no. 3, Paper No. S2R19, 21 pp., doi.org/10.54550/ECA2025V5S3R19. See arXiv2408.08421v4