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

1,3

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.

Links

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

Formula

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

Example

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

Prog

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

Crossrefs

Row sums of A305622.

Sequence in context: A297576 A164110 A285392 * A201351 A253470 A188899

Adjacent sequences: A327088 A327089 A327090 * A327092 A327093 A327094

Keyword

nonn

Author

Andrew Howroyd, Sep 13 2019

Status

approved