login
A371954
Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).
2
1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 5, 3, 0, 1, 0, 7, 0, 0, 0, 1, 0, 11, 6, 4, 0, 0, 1, 0, 15, 0, 0, 0, 0, 0, 1, 0, 22, 14, 0, 5, 0, 0, 0, 1, 0, 30, 0, 10, 0, 0, 0, 0, 0, 1, 0, 42, 25, 0, 0, 6, 0, 0, 0, 0, 1, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 77, 53, 30, 15, 0, 7, 0, 0, 0, 0, 0, 1
OFFSET
0,5
COMMENTS
A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.
EXAMPLE
Triangle begins:
1
0 1
0 2 1
0 3 0 1
0 5 3 0 1
0 7 0 0 0 1
0 11 6 4 0 0 1
0 15 0 0 0 0 0 1
0 22 14 0 5 0 0 0 1
0 30 0 10 0 0 0 0 0 1
0 42 25 0 0 6 0 0 0 0 1
0 56 0 0 0 0 0 0 0 0 0 1
0 77 53 30 15 0 7 0 0 0 0 0 1
Row n = 6 counts the following partitions:
. (6) (33) (222) . . (111111)
(51) (321) (2211)
(42) (3111) (21111)
(411) (2211) (111111)
(33) (21111)
(321) (111111)
(3111)
(222)
(2211)
(21111)
(111111)
MATHEMATICA
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Row n has A000005(n) positive entries.
Column k = 1 is A000041.
Column k = 2 is A002219 (aerated), ranks A357976.
Column k = 3 is A002220 (aerated), ranks A371955.
Removing all zeros gives A371783.
Row sums are A372121.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, complement A371796.
Sequence in context: A340867 A178616 A165252 * A127373 A200123 A187616
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Apr 20 2024
STATUS
approved