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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 introducedthe $d$-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 formulafor its Betti numbers and generalizing a result of Francisco, H\`a and Van Tuylon the cover ideal of Cohen-Macaulay bipartite graphs. Moreover we prove aBetti splitting result for the Alexander dual of a Cohen-Macaulay simplicialcomplex.

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