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A369924
Number of uniform words of length n with adjacent elements unequal using an infinite alphabet up to permutations of the alphabet.
2
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
0,5
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.
LINKS
FORMULA
a(n) = Sum_{d|n} A322013(d, n/d} for n > 0.
a(p) = 1 for prime p.
EXAMPLE
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}}.
PROG
(PARI) \\ Needs T(n, k) from A322013.
a(n) = {if(n==0, 1, sumdiv(n, d, T(d, n/d)))}
CROSSREFS
The case for adjacent elements possibly equal is A038041.
Cf. A322013, A369925 (circular words).
Sequence in context: A019426 A333599 A335405 * A347800 A341738 A128747
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Feb 06 2024
STATUS
approved