A316775 a(n) is the number of permutations of [1..n] that have the same number of inversions as non-inversions.

1, 1, 0, 0, 6, 22, 0, 0, 3836, 29228, 0, 0, 25598186, 296643390, 0, 0, 738680521142, 11501573822788, 0, 0, 62119523114983224, 1214967840930909302, 0, 0, 12140037056605135928410, 285899248139692651257566, 0, 0, 4759461354691529363949651814

Offset

0,5

Comments

a(n) is zero when n choose 2 is odd, that is for numbers that have remainders 2 or 3 when divided by 4.

Links

Table of n, a(n) for n=0..28.

Gal Beniamini, Nir Lavee, and Nati Linial, How Balanced Can Permutations Be?, arXiv:2306.16954 [math.CO], 2023. See p. 18.

Tanya Khovanova, 3-Symmetric Permutations

Wikipedia, Inversion

Formula

a(n) = A000140(n) if n in { A042948 }. - Alois P. Heinz, Oct 25 2018

Example

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.

Crossrefs

Cf. A000140, A042948.

Sequence in context: A261844 A372425 A007594 * A181593 A084539 A034124

Adjacent sequences: A316772 A316773 A316774 * A316776 A316777 A316778

Keyword

nonn

Author

Tanya Khovanova, Oct 22 2018

Extensions

a(10)-a(15) from Giovanni Resta, Oct 22 2018

a(16)-a(28) from Alois P. Heinz, Oct 24 2018

Status

approved