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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).

%I A371954 #8 Apr 20 2024 10:49:28
%S A371954 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,
%T A371954 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,
%U A371954 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
%N 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).
%C A371954 A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.
%e A371954 Triangle begins:
%e A371954   1
%e A371954   0  1
%e A371954   0  2  1
%e A371954   0  3  0  1
%e A371954   0  5  3  0  1
%e A371954   0  7  0  0  0  1
%e A371954   0 11  6  4  0  0  1
%e A371954   0 15  0  0  0  0  0  1
%e A371954   0 22 14  0  5  0  0  0  1
%e A371954   0 30  0 10  0  0  0  0  0  1
%e A371954   0 42 25  0  0  6  0  0  0  0  1
%e A371954   0 56  0  0  0  0  0  0  0  0  0  1
%e A371954   0 77 53 30 15  0  7  0  0  0  0  0  1
%e A371954 Row n = 6 counts the following partitions:
%e A371954   .  (6)       (33)      (222)     .  .  (111111)
%e A371954      (51)      (321)     (2211)
%e A371954      (42)      (3111)    (21111)
%e A371954      (411)     (2211)    (111111)
%e A371954      (33)      (21111)
%e A371954      (321)     (111111)
%e A371954      (3111)
%e A371954      (222)
%e A371954      (2211)
%e A371954      (21111)
%e A371954      (111111)
%t A371954 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A371954 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A371954 Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,0,10},{k,0,n}]
%Y A371954 Row n has A000005(n) positive entries.
%Y A371954 Column k = 1 is A000041.
%Y A371954 Column k = 2 is A002219 (aerated), ranks A357976.
%Y A371954 Column k = 3 is A002220 (aerated), ranks A371955.
%Y A371954 Removing all zeros gives A371783.
%Y A371954 Row sums are A372121.
%Y A371954 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371954 A321452 counts quanimous partitions, ranks A321454.
%Y A371954 A371789 counts non-quanimous sets, complement A371796.
%Y A371954 Cf. A002221, A002222, A006827, A035470, A064914, A237258, A321142, A321455, A365543, A371795.
%K A371954 nonn,tabl
%O A371954 0,5
%A A371954 _Gus Wiseman_, Apr 20 2024