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 modulo n.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{d|n} A369923(d, n/d} for n > 0.
a(p) = 1 for prime p.
EXAMPLE
a(1) = 0 because the symbol 'a' is considered to be adjacent to itself in a circular word. The set partition {{1}} is also excluded because 1 == 1 + 1 (mod 1).
The a(6) = 6 words are ababab, abacbc, abcabc, abcacb, abcbac, abcdef.
The corresponding a(6) = 6 set partitions are:
{{1,3,5},{2,4,6}},
{{1,3},{2,5},{4,6}},
{{1,4},{2,5},{3,6}},
{{1,4},{2,6},{3,5}},
{{1,5},{2,4},{3,6}},
{{1},{2},{3},{4},{5},{6}}.
PROG
(PARI) \\ Needs T(n, k) from A369923.
a(n) = {if(n==0, 1, sumdiv(n, d, T(d, n/d)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Feb 06 2024
STATUS
approved