A064914 -id:A064914 - cp's OEIS frontend

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).

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

Gus Wiseman, Apr 20 2024

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.

			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)

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.
Cf. A002221, A002222, A006827, A035470, A064914, A237258, A321142, A321455, A365543, 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,0,10},{k,0,n}]

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

Original entry on oeis.org

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

A372122 Number of strict triquanimous partitions of 3n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 5, 13, 18, 36, 51, 93, 132, 229, 315, 516, 735, 1134, 1575, 2407, 3309, 4878, 6710, 9690, 13168, 18744, 25114, 35050, 47210, 64503, 85573, 116445, 153328, 205367, 269383, 356668, 464268, 610644, 788274, 1026330, 1321017, 1704309, 2176054

Offset: 0

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums. Triquanimous partitions are counted by A002220 and ranked by A371955.

			The partition (11,7,5,4,3,2,1) has qualifying set partitions {{11},{4,7},{1,2,3,5}} and {{11},{1,3,7},{2,4,5}} so is counted under a(11).
The a(5) = 1 through a(9) = 13 partitions:
  (5,4,3,2,1)  (6,5,4,2,1)  (7,5,4,3,2)    (8,6,5,3,2)    (9,6,5,4,3)
                            (7,6,4,3,1)    (8,7,5,3,1)    (9,7,5,4,2)
                            (7,6,5,2,1)    (8,7,6,2,1)    (9,7,6,3,2)
                            (6,5,4,3,2,1)  (7,6,5,3,2,1)  (9,8,5,4,1)
                                           (8,6,4,3,2,1)  (9,8,6,3,1)
                                                          (9,8,7,2,1)
                                                          (7,6,5,4,3,2)
                                                          (8,6,5,4,3,1)
                                                          (8,7,5,4,2,1)
                                                          (8,7,6,3,2,1)
                                                          (9,6,5,4,2,1)
                                                          (9,7,5,3,2,1)
                                                          (9,8,4,3,2,1)

The non-strict biquanimous version is A002219, ranks A357976.
The non-strict version is A002220, ranks A371955.
The biquanimous version is A237258, ranks A357854.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454, strict A371737.
A371783 counts k-quanimous partitions.
A371795 counts non-biquanimous partitions, even case A006827, ranks A371731.
Cf. A035470, A064914, A321142, A321455, A371781, A371796.

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[3n], UnsameQ@@#&&Select[facs[Times@@Prime/@#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 30 2025

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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).

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

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A372122 Number of strict triquanimous partitions of 3n.

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