A371954 -id:A371954 - cp's OEIS frontend

A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

1, 3, 4, 9, 8, 22, 16, 42, 41, 74, 57, 183, 102, 233, 263, 463, 298, 875, 491, 1350, 1172, 1775, 1256, 4273, 2225, 4399, 4584, 8049, 4566, 14913, 6843, 18539, 15831, 22894, 18196, 53323, 21638, 48947, 50281, 94500, 44584, 144976, 63262, 173436, 169361, 202153

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums. The triangles A371783 and A371954 count k-quanimous partitions.

Row sums of A371783.
Row sums of A371954.
A000005 counts divisors.
A000041 counts integer partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A321452 counts quanimous partitions, complement A321451.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A035470, A064914, A321455, A365543, A371737, A371791, A371795.

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[Sum[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{k,Divisors[n]}],{n,1,10}]

T(n, d) = my(v=partitions(n/d), w=List([])); forvec(s=vector(d, i, [1, #v]), listput(w, vecsort(concat(vector(d, i, v[s[i]])))), 1); #Set(w);
a(n) = sumdiv(n, d, T(n, d)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

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.

Original entry on oeis.org

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, 77, 53, 30, 15, 7, 1, 101, 1, 135, 89, 8, 1, 176, 65, 21, 1, 231, 167, 55, 9, 1, 297, 1, 385, 278, 173, 28, 10, 1, 490, 1, 627, 480, 140, 91, 11, 1, 792, 343, 36, 1

Offset: 1

Gus Wiseman, Apr 14 2024

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

			Triangle begins:
    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
   77  53  30  15   7   1
  101   1
  135  89   8   1
  176  65  21   1
Row n = 6 counts the following partitions:
  (6)       (33)      (222)     (111111)
  (33)      (321)     (2211)
  (42)      (2211)    (21111)
  (51)      (3111)    (111111)
  (222)     (21111)
  (321)     (111111)
  (411)
  (2211)
  (3111)
  (21111)
  (111111)

Row lengths are A000005.
Column k = 1 is A000041.
Inserting zeros gives A371954.
Row sums are A372121.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371736 counts non-quanimous strict partitons, complement A371737.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A027750, A035470, A064914, A321455, A365543, A371791, A371795.

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,1,8},{k,Divisors[n]}]

More terms from Jinyuan Wang, Feb 13 2025

Name edited by Peter Munn, Mar 05 2025

cp's OEIS Frontend

A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions