Sparre Andersen's number of positives of a signed permutation.
For $\pi$ a signed permutation of length $n$, first create the tuple $x = (x_1, \dots, x_n)$, where $x_i = c_{|\pi_1|} \operatorname{sgn}(\pi_{|\pi_1|}) + \cdots + c_{|\pi_i|} \operatorname{sgn}(\pi_{|\pi_i|})$ and $(c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1})$. The actual value of the c-tuple for Andersen's statistic does not matter so long as no sums or differences of any subset of the $c_i$'s is zero. The choice of powers of $2$ is just a convenient choice.
This returns the number of strictly positive values in the $x$-tuple. This is related to the discrete arcsin distribution. The number of signed permutations with value equal to $k$ is given by $\binom{2k}{k} \binom{2n-2k}{n-k} \frac{n!}{2^n}$. This statistic is equidistributed with Sparre Andersen's `Position of Maximum' statistic.

Return the signed permutation with all signs reversed.

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

The signed permutation with all signs positive.