cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A382077 Number of integer partitions of n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 171, 217, 283, 361, 449, 574, 721, 900, 1126, 1397, 1731, 2143, 2632, 3223, 3961, 4825, 5874, 7131, 8646, 10452, 12604, 15155, 18216, 21826, 26108, 31169, 37156, 44202, 52492, 62233, 73676, 87089, 102756, 121074
Offset: 0

Views

Author

Gus Wiseman, Mar 18 2025

Keywords

Comments

First differs from A240306 at a(14) = 76, A240306(14) = 77.
First differs from A381992 at a(17) = 171, A381992(17) = 170.

Examples

			For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

Crossrefs

Factorizations of this type are counted by A050345.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions of this type are counted by A116539.
The MM-numbers of these multiset partitions are A302494.
Twice-partitions of this type are counted by A358914.
For distinct block-sums instead of blocks we have A381992, ranked by A382075.
The complement is counted by A382078, unique A382079.
These partitions are ranked by A382200, complement A293243.
For normal multisets instead of integer partitions we have A382214, complement A292432.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A292444, A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870, A381990.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]

Extensions

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A240309 Number of partitions p of n such that (maximal multiplicity of the parts of p) > (number of distinct parts of p).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 59, 76, 100, 128, 173, 212, 275, 350, 447, 552, 704, 870, 1094, 1346, 1672, 2048, 2540, 3084, 3775, 4595, 5592, 6764, 8180, 9853, 11865, 14250, 17075, 20404, 24376, 29024, 34498, 41012, 48550, 57463, 67873
Offset: 0

Views

Author

Clark Kimberling, Apr 05 2014

Keywords

Examples

			a(6) counts these 5 partitions:  33, 3111, 222, 21111, 111111.

Crossrefs

Cf. A240305, A240306, A239964, A240308, A000041.

Programs

  • Mathematica
    z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

Formula

a(n) = A240308(n) - A239964(n) for n >= 0.
a(n) + A240305(n) + A239964(n) = A000041(n) for n >= 0.

A240305 Number of partitions p of n such that (maximal multiplicity of the parts of p) < (number of distinct parts of p).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 5, 7, 12, 14, 21, 29, 38, 50, 70, 90, 117, 156, 196, 253, 324, 411, 514, 650, 809, 1015, 1259, 1555, 1917, 2365, 2898, 3536, 4318, 5248, 6365, 7691, 9297, 11180, 13446, 16115, 19296, 23019, 27474, 32653, 38838, 46002, 54511, 64371, 76012
Offset: 0

Views

Author

Clark Kimberling, Apr 05 2014

Keywords

Examples

			a(6) counts these 3 partitions:  51, 42, 321.

Crossrefs

Cf. A240306, A239964, A240308, A240309, A000041.

Programs

  • Mathematica
    z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

Formula

a(n) = A240306(n) - A239964(n) for n >= 0.
a(n) + A239964(n) + A240309(n) = A000041(n) for n >= 0.

A240308 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 8, 10, 15, 18, 28, 35, 48, 63, 85, 106, 141, 180, 229, 294, 374, 468, 591, 741, 925, 1149, 1421, 1751, 2163, 2648, 3239, 3944, 4813, 5825, 7062, 8518, 10286, 12340, 14835, 17739, 21223, 25287, 30155, 35787, 42522, 50296, 59556, 70243, 82902
Offset: 0

Views

Author

Clark Kimberling, Apr 05 2014

Keywords

Examples

			a(6) counts these 8 partitions:  6, 411, 33, 3111, 222, 2211, 21111, 111111.

Crossrefs

Cf. A240305, A240306, A239964, A240309, A000041.

Programs

  • Mathematica
    z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

Formula

a(n) = A239964(n) + A240308(n) for n >= 0.
a(n) + A240305(n) = A000041(n) for n >= 0.
Showing 1-4 of 4 results.

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