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let $\mathcal{ABP}$, the set of "asymmetric, balanced permutation matrices", be defined as the set of permutation matrices, that aren't equal their transpose but, for which the number $1$s below the principal diagonal equals the number of $1$s above that diagonal, i.e.: $P\in\mathcal{ABP}\implies$ $$p_{ij}\in\{0,1\}\\ \sum_{i=1}^{n}p_{ij}=\sum_{j=1}^{n}p_{ij}=1\\ \sum_{i<j}p_{ij}=\sum_{j<i}p_{ij}\\ P\ne P^T$$

Question:
how many asymmetric, balanced permutation matrices exist for given dimension $n$?

(other information about those matrices or permutation is also appreciated.)

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    $\begingroup$ Note that all involutions are known (these are symmetric, and thus also balanced). You can then compute only balanced permutations, and then take the difference. The number of balanced permutations seems to be given by oeis.org/A010551 after a computer experiment... $\endgroup$
    – Per Alexandersson
    Commented Jun 6, 2016 at 15:15
  • $\begingroup$ @PerAlexandersson after some ruminating I believe that the balanced permutation matrices should correspond to permutations, for which the number of positive differences of adjacent elements equals that of the negative differences and, after some searching I found out that the number of such permutations are the Kendall-Mann numbers; they correspond to oeis.org/A000140 and have been the subject of this question. Sadly, I only found a recursive function for involutions. $\endgroup$
    – Manfred Weis
    Commented Jun 14, 2016 at 17:06

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