A275714 Number T(n,k) of set partitions of [n] into k blocks with equal element sum; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.
1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 0, 1, 7, 3, 1, 0, 1, 0, 9, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 35, 43, 0, 0, 1, 0, 1, 62, 102, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 595, 0, 68, 0, 1, 0, 1, 361, 1480, 871, 187, 17, 0, 1
Offset: 0
Examples
T(8,1) = 1: 12345678. T(8,2) = 7: 12348|567, 12357|468, 12456|378, 1278|3456, 1368|2457, 1458|2367, 1467|2358. T(8,3) = 3: 1236|48|57, 138|246|57, 156|237|48. T(8,4) = 1: 18|27|36|45. T(9,3) = 9: 12345|69|78, 1239|456|78, 1248|357|69, 1257|348|69, 1347|258|69, 1356|249|78, 159|2346|78, 168|249|357, 159|267|348. Triangle T(n,k) begins: 00 : 1; 01 : 0, 1; 02 : 0, 1; 03 : 0, 1, 1; 04 : 0, 1, 1; 05 : 0, 1, 0, 1; 06 : 0, 1, 0, 1; 07 : 0, 1, 4, 0, 1; 08 : 0, 1, 7, 3, 1; 09 : 0, 1, 0, 9, 0, 1; 10 : 0, 1, 0, 0, 0, 1; 11 : 0, 1, 35, 43, 0, 0, 1; 12 : 0, 1, 62, 102, 0, 0, 1; 13 : 0, 1, 0, 0, 0, 0, 0, 1; 14 : 0, 1, 0, 595, 0, 68, 0, 1; 15 : 0, 1, 361, 1480, 871, 187, 17, 0, 1;
Links
- Alois P. Heinz, Rows n = 0..34, flattened
- Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
- Wikipedia, Partition of a set
Crossrefs
Programs
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Mathematica
Needs["Combinatorica`"]; T[n_, k_] := Count[(Equal @@ (Total /@ #)&) /@ KSetPartitions[n, k], True]; Table[row = Table[T[n, k], {k, 0, Ceiling[n/2]}]; Print[n, " ", row]; row, {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 20 2017 *)