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

A natural extension of bipartite graphs are d-partite clutters, where d >= 2 is an integer. For a poset P, Ene, Herzog and Mohammadi introduced the d-partite clutter C_{P,d} of multichains of length d in P, showing that it is Cohen-Macaulay. We prove that the cover ideal of C_{P,d} admits an x(i)-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, Ha 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. Interesting examples are given, in particular the first example of ideal that does not admit Betti splitting in any characteristic.