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

Author

Alois P. Heinz, Aug 06 2016

Keywords

• nonn
• tabf

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

Columns k=0-5 give: A000007, A000012 (for n>0), A058377, A112972, A317806, A317807.

Row sums give A035470 = 1 + A112956.

T(n^2,n) gives A321282.

Cf. A248112.

Programs

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 *)