Recursive Betti numbers for Cohen-Macaulay $d$-partite clutters arising from posets

A natural extension of bipartite graphs are $d$-partite clutters, where $d \geq 2$isaninteger\mathrm{.}Foraposet$P$, Ene, Herzog and Mohammadi introduced the $d$-partite clutter ${\mathcal{C}}_{P,d}$ of multichains of length $d$ in $P$, showing that it is Cohen-Macaulay. We prove that the cover ideal of ${\mathcal{C}}_{P,d}$ admits an $x_i$-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, H`a and Van Tuyl on the cover ideal of Cohen-Macaulay bipartite graphs. Moreover we prove a Betti splitting result for the Alexander dual of a Cohen-Macaulay simplicial complex.