%I #9 Apr 20 2024 10:49:25

%S 1,2,1,3,1,5,3,1,7,1,11,6,4,1,15,1,22,14,5,1,30,10,1,42,25,6,1,56,1,

%T 77,53,30,15,7,1,101,1,135,89,8,1,176,65,21,1

%N Irregular triangle read by rows where T(n,d) is the number of integer partitions of n that can be partitioned into d blocks with equal sums, with d ranging over all divisors d|n.

%C These could be called d-quanimous partitions, cf. A002219, A064914, A321452.

%e Triangle begins:

%e 1

%e 2 1

%e 3 1

%e 5 3 1

%e 7 1

%e 11 6 4 1

%e 15 1

%e 22 14 5 1

%e 30 10 1

%e 42 25 6 1

%e 56 1

%e 77 53 30 15 7 1

%e 101 1

%e 135 89 8 1

%e 176 65 21 1

%e Row n = 6 counts the following partitions:

%e (6) (33) (222) (111111)

%e (33) (321) (2211)

%e (42) (2211) (21111)

%e (51) (3111) (111111)

%e (222) (21111)

%e (321) (111111)

%e (411)

%e (2211)

%e (3111)

%e (21111)

%e (111111)

%t hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];

%t facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];

%t Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,1,8},{k,Divisors[n]}]

%Y Row lengths are A000005.

%Y Column k = 1 is A000041.

%Y Inserting zeros gives A371954.

%Y Row sums are A372121.

%Y A002219 (aerated) counts biquanimous partitions, ranks A357976.

%Y A237258 aerated counts biquanimous strict partitions, ranks A357854.

%Y A321142 and A371794 count non-biquanimous strict partitions.

%Y A321451 counts non-quanimous partitions, ranks A321453.

%Y A321452 counts quanimous partitions, ranks A321454.

%Y A371736 counts non-quanimous strict partitons, complement A371737.

%Y A371781 lists numbers with biquanimous prime signature, complement A371782.

%Y A371789 counts non-quanimous sets, differences A371790.

%Y A371796 counts quanimous sets, differences A371797.

%Y Cf. A006827, A035470, A064914, A321455, A365543, A371791, A371795.

%K nonn,tabf,more

%O 1,2

%A _Gus Wiseman_, Apr 14 2024