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A364285 Number T(n,k) of partitions of n with largest part k where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 4
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 4, 1, 0, 1, 15, 20, 5, 1, 0, 1, 31, 81, 30, 6, 1, 0, 1, 63, 287, 175, 42, 7, 1, 0, 1, 127, 952, 841, 280, 56, 8, 1, 0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1, 0, 1, 511, 9370, 23820, 10333, 2730, 600, 90, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.
T(n,k) is also the number of endofunctions on [n] such that k is the range maximum and the number of elements that are mapped to m is divisible by m.
T(4,2) = 7: (2211), (2121), (2112), (1221), (1212), (1122), (2222).
T(5,3) = 20: (33322), (33232), (33223), (32332), (32323), (32233), (23332), (23323), (23233), (22333), (33311), (33131), (33113), (31331), (31313), (31133), (13331), (13313), (13133), (11333).
LINKS
EXAMPLE
T(4,1) = 1: 1111abcd.
T(4,2) = 7: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab, 22abcd.
T(4,3) = 4: 3abc1d, 3abd1c, 3acd1b, 3bcd1a.
T(4,4) = 1: 4abcd.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 7, 4, 1;
0, 1, 15, 20, 5, 1;
0, 1, 31, 81, 30, 6, 1;
0, 1, 63, 287, 175, 42, 7, 1;
0, 1, 127, 952, 841, 280, 56, 8, 1;
0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
end:
T:= (n, k)-> b(n, k)-`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
CROSSREFS
Columns k=0-2 give: A000007, A057427, A000225(n-1).
Row sums give A178682.
T(2n,n) gives A364322.
Cf. A364310.
Sequence in context: A227320 A318507 A055807 * A213060 A272008 A054024
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 17 2023
STATUS
approved

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Last modified June 25 15:24 EDT 2024. Contains 373705 sequences. (Running on oeis4.)