A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 5, 16, 15, 1, 1, 1, 9, 64, 131, 52, 1, 1, 1, 17, 298, 1613, 1496, 203, 1, 1, 1, 33, 1540, 25097, 69026, 22482, 877, 1, 1, 1, 65, 8506, 461105, 4383626, 4566992, 426833, 4140, 1, 1, 1, 129, 48844, 9483041, 350813126, 1394519922, 437665649, 9934563, 21147, 1
Offset: 0
Examples
A(2,2) = 3: 1234, 12|34, 14|23. A(2,3) = 5: 123456, 123|456, 126|345, 135|246, 156|234. A(2,4) = 9: 12345678, 1234|5678, 1238|4567, 1247|3568, 1278|3456, 1346|2578, 1368|2457, 1467|2358, 1678|2345. A(3,2) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 5, 9, 17, 33, ... 1, 5, 16, 64, 298, 1540, 8506, ... 1, 15, 131, 1613, 25097, 461105, 9483041, ... 1, 52, 1496, 69026, 4383626, 350813126, 33056715626, ... 1, 203, 22482, 4566992, 1394519922, 573843627152, 293327384637282, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..60, flattened
- J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
A:= proc(n, k) option remember; `if`(k*n=0, 1, add( binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n) end: seq(seq(A(n, d-n), n=0..d), d=0..12); -
Mathematica
A[n_, k_] := A[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*A[j, k], {j, 0, n-1}]/n]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)