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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.
17

%I #26 Aug 16 2021 13:55:14

%S 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,

%T 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,

%U 595,0,68,0,1,0,1,361,1480,871,187,17,0,1

%N 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.

%H Alois P. Heinz, <a href="/A275714/b275714.txt">Rows n = 0..34, flattened</a>

%H Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/s11139-021-00418-7">On k-partitions of multisets with equal sums</a>, The Ramanujan J. (2021) Vol. 55, 421-435.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e T(8,1) = 1: 12345678.

%e T(8,2) = 7: 12348|567, 12357|468, 12456|378, 1278|3456, 1368|2457, 1458|2367, 1467|2358.

%e T(8,3) = 3: 1236|48|57, 138|246|57, 156|237|48.

%e T(8,4) = 1: 18|27|36|45.

%e 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.

%e Triangle T(n,k) begins:

%e 00 : 1;

%e 01 : 0, 1;

%e 02 : 0, 1;

%e 03 : 0, 1, 1;

%e 04 : 0, 1, 1;

%e 05 : 0, 1, 0, 1;

%e 06 : 0, 1, 0, 1;

%e 07 : 0, 1, 4, 0, 1;

%e 08 : 0, 1, 7, 3, 1;

%e 09 : 0, 1, 0, 9, 0, 1;

%e 10 : 0, 1, 0, 0, 0, 1;

%e 11 : 0, 1, 35, 43, 0, 0, 1;

%e 12 : 0, 1, 62, 102, 0, 0, 1;

%e 13 : 0, 1, 0, 0, 0, 0, 0, 1;

%e 14 : 0, 1, 0, 595, 0, 68, 0, 1;

%e 15 : 0, 1, 361, 1480, 871, 187, 17, 0, 1;

%t 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 *)

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

%Y Row sums give A035470 = 1 + A112956.

%Y T(n^2,n) gives A321282.

%Y Cf. A248112.

%K nonn,tabf

%O 0,22

%A _Alois P. Heinz_, Aug 06 2016