Segre powers of posets preserve EL-shellability

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 Bj\"orner 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].