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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.

%I A275714 #26 Aug 16 2021 13:55:14
%S A275714 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 A275714 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 A275714 595,0,68,0,1,0,1,361,1480,871,187,17,0,1
%N 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.
%H A275714 Alois P. Heinz, <a href="/A275714/b275714.txt">Rows n = 0..34, flattened</a>
%H A275714 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 A275714 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A275714 T(8,1) = 1: 12345678.
%e A275714 T(8,2) = 7: 12348|567, 12357|468, 12456|378, 1278|3456, 1368|2457, 1458|2367, 1467|2358.
%e A275714 T(8,3) = 3: 1236|48|57, 138|246|57, 156|237|48.
%e A275714 T(8,4) = 1: 18|27|36|45.
%e A275714 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 A275714 Triangle T(n,k) begins:
%e A275714 00 :  1;
%e A275714 01 :  0,  1;
%e A275714 02 :  0,  1;
%e A275714 03 :  0,  1,   1;
%e A275714 04 :  0,  1,   1;
%e A275714 05 :  0,  1,   0,    1;
%e A275714 06 :  0,  1,   0,    1;
%e A275714 07 :  0,  1,   4,    0,   1;
%e A275714 08 :  0,  1,   7,    3,   1;
%e A275714 09 :  0,  1,   0,    9,   0,   1;
%e A275714 10 :  0,  1,   0,    0,   0,   1;
%e A275714 11 :  0,  1,  35,   43,   0,   0,  1;
%e A275714 12 :  0,  1,  62,  102,   0,   0,  1;
%e A275714 13 :  0,  1,   0,    0,   0,   0,  0, 1;
%e A275714 14 :  0,  1,   0,  595,   0,  68,  0, 1;
%e A275714 15 :  0,  1, 361, 1480, 871, 187, 17, 0, 1;
%t A275714 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 A275714 Columns k=0-5 give: A000007, A000012 (for n>0), A058377, A112972, A317806, A317807.
%Y A275714 Row sums give A035470 = 1 + A112956.
%Y A275714 T(n^2,n) gives A321282.
%Y A275714 Cf. A248112.
%K A275714 nonn,tabf
%O A275714 0,22
%A A275714 _Alois P. Heinz_, Aug 06 2016