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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.
17
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
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
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: A120324 A136630 A387908 * A111728 A359010 A272774
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Aug 06 2016
STATUS
approved