A369925 -id:A369925 - cp's OEIS frontend

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

Andrew Howroyd, Feb 06 2024

nonn

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.

			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}}.

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

The case for adjacent elements possibly equal is A038041.

Cf. A322013, A369925 (circular words).

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

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

a(p) = 1 for prime p.

cp's OEIS Frontend

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

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