A327091 Number of chiral pairs of length n words with integer entries that cover an initial interval of positive integers.

0, 1, 5, 36, 264, 2335, 23609, 272880, 3543360, 51123511, 811313945, 14045781456, 263429150544, 5320671461575, 115141595216009, 2657827340717760, 65185383511024320, 1692767331624879031, 46400793659613081785, 1338843898122140977776, 40562412499251225624624

Offset: 1

Author

Andrew Howroyd, Sep 13 2019

Keywords

• nonn

Comments

If the word is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			a(3) = 5 because there are the following chiral pairs of words:
  112/211, 122/221,
  123/321, 132/231, 213/312.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Row sums of A305622.

Programs

PARI

a(n) = {sum(k=1, n, k! * (stirling(n, k, 2) - stirling((n+1)\2, k, 2)) / 2)}

Formula

a(n) = Sum_{k=1..n} (k!/2) * (Stirling2(n, k) - Stirling2(ceiling(n/2), k)).