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

A natural extension of bipartite graphs are d  -partite clutters, where d≥2d≥2 is an integer. For a poset P, Ene, Herzog and Mohammadi introduced the d  -partite clutter CP,dCP,d of multichains of length d in P  , showing that it is Cohen–Macaulay. We prove that the cover ideal of CP,dCP,d admits an xixi-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, Hà 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.