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A316775 a(n) is the number of permutations of [1..n] that have the same number of inversions as non-inversions. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A190275 A261844 A007594 * A181593 A084539 A034124
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

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Last modified February 25 06:40 EST 2024. Contains 370310 sequences. (Running on oeis4.)