A369924 Number of uniform words of length n with adjacent elements unequal using an infinite alphabet up to permutations of the alphabet.

1, 1, 1, 1, 2, 1, 7, 1, 38, 30, 331, 1, 5560, 1, 47846, 164585, 815693, 1, 35149698, 1, 338596631, 4420377702, 4939227217, 1, 1430570927009, 66218360626, 2850860253242, 372419004321831, 628358300200811, 1, 156433852692766134, 1, 2606291948338277064

Offset: 0

Author

Andrew Howroyd, Feb 06 2024

Keywords

• nonn

Comments

A word is uniform here if each symbol that occurs in the word occurs with the same frequency.

a(n) is the number of ways to partition [n] into parts of equal size and no part containing values that differ by 1.

Examples

			The a(4) = 2 words are abab, abcd.
The a(6) = 7 words are ababab, abacbc, abcabc, abcacb, abcbac, abcbca, abcdef.
The a(4) = 2 set partitions are {{1,3}, {2,4}} and {{1},{2},{3},{4}}.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

The case for adjacent elements possibly equal is A038041.

Cf. A322013, A369925 (circular words).

Programs

PARI

\\ Needs T(n,k) from A322013.
a(n) = {if(n==0, 1, sumdiv(n, d, T(d, n/d)))}

Formula

a(n) = Sum_{d|n} A322013(d, n/d) for n > 0.

a(p) = 1 for prime p.