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A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.

%I A371783 #16 Mar 05 2025 22:05:02
%S A371783 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 A371783 77,53,30,15,7,1,101,1,135,89,8,1,176,65,21,1,231,167,55,9,1,297,1,
%U A371783 385,278,173,28,10,1,490,1,627,480,140,91,11,1,792,343,36,1
%N A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.
%C A371783 These could be called d-quanimous partitions, cf. A002219, A064914, A321452.
%e A371783 Triangle begins:
%e A371783     1
%e A371783     2   1
%e A371783     3   1
%e A371783     5   3   1
%e A371783     7   1
%e A371783    11   6   4   1
%e A371783    15   1
%e A371783    22  14   5   1
%e A371783    30  10   1
%e A371783    42  25   6   1
%e A371783    56   1
%e A371783    77  53  30  15   7   1
%e A371783   101   1
%e A371783   135  89   8   1
%e A371783   176  65  21   1
%e A371783 Row n = 6 counts the following partitions:
%e A371783   (6)       (33)      (222)     (111111)
%e A371783   (33)      (321)     (2211)
%e A371783   (42)      (2211)    (21111)
%e A371783   (51)      (3111)    (111111)
%e A371783   (222)     (21111)
%e A371783   (321)     (111111)
%e A371783   (411)
%e A371783   (2211)
%e A371783   (3111)
%e A371783   (21111)
%e A371783   (111111)
%t A371783 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A371783 facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t A371783 Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,1,8},{k,Divisors[n]}]
%Y A371783 Row lengths are A000005.
%Y A371783 Column k = 1 is A000041.
%Y A371783 Inserting zeros gives A371954.
%Y A371783 Row sums are A372121.
%Y A371783 A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A371783 A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A371783 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371783 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371783 A321452 counts quanimous partitions, ranks A321454.
%Y A371783 A371736 counts non-quanimous strict partitons, complement A371737.
%Y A371783 A371781 lists numbers with biquanimous prime signature, complement A371782.
%Y A371783 A371789 counts non-quanimous sets, differences A371790.
%Y A371783 A371796 counts quanimous sets, differences A371797.
%Y A371783 Cf. A006827, A027750, A035470, A064914, A321455, A365543, A371791, A371795.
%K A371783 nonn,tabf
%O A371783 1,2
%A A371783 _Gus Wiseman_, Apr 14 2024
%E A371783 More terms from _Jinyuan Wang_, Feb 13 2025
%E A371783 Name edited by _Peter Munn_, Mar 05 2025