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%I #32 Jul 06 2023 01:58:18
%S 1,1,0,0,6,22,0,0,3836,29228,0,0,25598186,296643390,0,0,738680521142,
%T 11501573822788,0,0,62119523114983224,1214967840930909302,0,0,
%U 12140037056605135928410,285899248139692651257566,0,0,4759461354691529363949651814
%N a(n) is the number of permutations of [1..n] that have the same number of inversions as non-inversions.
%C a(n) is zero when n choose 2 is odd, that is for numbers that have remainders 2 or 3 when divided by 4.
%H Gal Beniamini, Nir Lavee, and Nati Linial, <a href="https://arxiv.org/abs/2306.16954">How Balanced Can Permutations Be?</a>, arXiv:2306.16954 [math.CO], 2023. See p. 18.
%H Tanya Khovanova, <a href="https://blog.tanyakhovanova.com/2018/10/3-symmetric-permutations/#comment-12716">3-Symmetric Permutations</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion</a>
%F a(n) = A000140(n) if n in { A042948 }. - _Alois P. Heinz_, Oct 25 2018
%e Consider a permutation 1432. It has exactly three pairs of numbers, the first of them is 1, that are in increasing order. The other three pairs are in decreasing order. The other 5 permutations of size 4 with this property are 2341, 2413, 3142, 3214, 4123. Thus a(4) = 6.
%Y Cf. A000140, A042948.
%K nonn
%O 0,5
%A _Tanya Khovanova_, Oct 22 2018
%E a(10)-a(15) from _Giovanni Resta_, Oct 22 2018
%E a(16)-a(28) from _Alois P. Heinz_, Oct 24 2018