|
| |
|
|
A275422
|
|
Number A(n,k) of set partitions of [n] such that k is a multiple of each block size; square array A(n,k), n>=0, k>=0, read by antidiagonals.
|
|
10
|
|
|
|
1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 15, 1, 1, 1, 4, 1, 52, 1, 1, 2, 2, 10, 1, 203, 1, 1, 1, 4, 5, 26, 1, 877, 1, 1, 2, 1, 11, 11, 76, 1, 4140, 1, 1, 1, 5, 1, 31, 31, 232, 1, 21147, 1, 1, 2, 1, 14, 2, 106, 106, 764, 1, 115975, 1, 1, 1, 4, 1, 46, 7, 372, 337, 2620, 1, 678570
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,6
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. for column k>0: exp(Sum_{d|k} x^d/d!), for k=0: exp(exp(x)-1).
|
|
|
EXAMPLE
|
A(5,3) = 11: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345, 1|2|3|4|5.
A(4,4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
A(6,5) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
: 2, 1, 2, 1, 2, 1, 2, 1, 2, ...
: 5, 1, 4, 2, 4, 1, 5, 1, 4, ...
: 15, 1, 10, 5, 11, 1, 14, 1, 11, ...
: 52, 1, 26, 11, 31, 2, 46, 1, 31, ...
: 203, 1, 76, 31, 106, 7, 167, 1, 106, ...
: 877, 1, 232, 106, 372, 22, 659, 2, 372, ...
: 4140, 1, 764, 337, 1499, 57, 2836, 9, 1500, ...
|
|
|
MAPLE
|
A:= proc(n, k) option remember; `if`(n=0, 1, add(
`if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=
`if`(k=0, 1..n, numtheory[divisors](k))))
end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
|
|
|
MATHEMATICA
|
A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[j>n, 0, A[n-j, k]*Binomial[n-1, j - 1]], {j, If[k==0, Range[n], Divisors[k]]}]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
|
|
|
CROSSREFS
|
Columns k=0-10 give: A000110, A000012, A000085, A190865, A190452, A275423, A275424, A275425, A275426, A275427, A275428.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
