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

A natural extension of bipartite graphs are $d$-partite clutters, where $d \geq 2$ is an integer. For a poset $P$, 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.