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A369925
Number of uniform circular words of length n with adjacent elements unequal using an infinite alphabet up to permutations of the alphabet.
2
1, 0, 1, 1, 2, 1, 6, 1, 33, 23, 295, 1, 4877, 1, 44191, 141210, 749316, 1, 31762349, 1, 309754506, 3980911205, 4704612121, 1, 1303743206944, 55279816357, 2737023412201, 343866841144704, 564548508168226, 1, 145630899385513158, 1, 2359434158555273239
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
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
The case for adjacent elements possibly equal is A038041.
Cf. A369923, A369924 (linear words).
Sequence in context: A277440 A321597 A083720 * A055878 A331654 A346864
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Feb 06 2024
STATUS
approved