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Let n be a positive integer, let sigma be a noncrossing partition on {1,2,...,2n}. Consider the set OuterMax(sigma) := {max(W); W is an outer block of sigma}.

We say that sigma is anti-commutator friendly when it satisfies:

1) OuterMax(sigma) is a subset of the union of {1,3,...,2n-1} and {2n}.

2) For every j in {1, 3,...,2n-1} \ OuterMax(sigma), one has depth(j) is not equal to depth(j + 1), where depth(j) stands for the depth of the block of sigma which contains the number j.

For example for n=1 only partition {{1},{2}} is anti-commutator friendly.

Multiplied by two, gives sequence of Boolean cumulants of ab+ba, for a,b freely independent both distributed (delta_0+delta_2)/2. See M. Fevrier et al. Proposition 6.11.