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

0,22

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

Example

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;

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 *)

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.

Sequence in context: A036861 A120324 A136630 * A111728 A359010 A143784

Adjacent sequences: A275711 A275712 A275713 * A275715 A275716 A275717

Keyword

nonn,tabf

Author

Alois P. Heinz, Aug 06 2016

Status

approved