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.

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A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9
Offset: 0

Views

Author

Gus Wiseman, Mar 30 2025

Keywords

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

Examples

			The normal multiset {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}

Crossrefs

For constant instead of strict blocks we have A000045.
Factorizations of this type are counted by A050326, with distinct sums A381633.
For the strong case see A292444, A382430, complement A381996, A382523.
MM-numbers of sets of sets are A302494, see A302478, A382201.
Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.
For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).
With distinct sums we have A382459.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.

Programs

  • Mathematica
    allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n], Length[Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]

A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 2, 7, 4, 10, 19
Offset: 0

Views

Author

Gus Wiseman, Apr 01 2025

Keywords

Comments

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

Examples

			The normal multiset {1,2,2,2,2,3,3,4} has only one multiset partition into a set of sets with distinct sums: {{2},{1,2},{2,3},{2,3,4}}, so is counted under a(8).
The a(1) = 1 through a(7) = 7 multisets:
  {1}  .  {112}  {1122}  {11123}  {111233}  {1111234}
          {122}          {12223}  {122233}  {1112223}
                         {12333}            {1112333}
                                            {1222234}
                                            {1222333}
                                            {1233334}
                                            {1234444}

Crossrefs

Twice-partitions of this type are counted by A279785, A270995, A358914.
Factorizations of this type are counted by A381633, A050320, A050326.
Normal multiset partitions of this type are A381718, A116540, A116539.
Multiset partitions of this type are ranked by A382201, A302478, A302494.
For at least one choice: A382216 (strict A382214), complement A382202 (strict A292432).
For the strong case see: A382430 (strict A292444), complement A382523 (strict A381996).
Without distinct sums we have A382458.
For integer partitions we have A382460, ranks A381870, strict A382079, ranks A293511.
Set multipartitions: A089259, A296119, A318360.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A317532, A321469, A326519, A381078, A381441, A382428.

Programs

  • Mathematica
    allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]]==1&]],{n,0,5}]

A255942 Number of collections of nonempty multisets with a total of n objects of exactly two colors.

Original entry on oeis.org

2, 8, 23, 56, 127, 268, 547, 1072, 2055, 3836, 7045, 12692, 22566, 39542, 68520, 117354, 199082, 334446, 557175, 920528, 1509606, 2457698, 3974813, 6387026, 10201990, 16201424, 25589505, 40205766, 62857564, 97800674, 151475564, 233575222, 358656011, 548476318
Offset: 2

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

Column k=2 of A255903.

Formula

a(n) = A255903(n,2).

A255951 Number of collections of nonempty multisets with a total of n+1 objects of exactly n colors.

Original entry on oeis.org

0, 2, 8, 33, 144, 675, 3396, 18270, 104656, 635679, 4078460, 27544781, 195219624, 1447922814, 11210516128, 90405000105, 757823763968, 6591085453935, 59379154929180, 553270283842586, 5324267165971800, 52849751984814183, 540469505506572620, 5688102295215875017
Offset: 0

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

A diagonal of A255903.

Formula

a(n) = A255903(n+1,n).

A255907 Number of collections of nonempty multisets with a total of 2n objects of exactly n colors.

Original entry on oeis.org

1, 2, 23, 492, 15800, 693667, 39371078, 2780643510, 237625842970, 24048952078889, 2833749745048664, 383425910861623475, 58894162142705623291, 10170230140575830362828, 1958227054977871232165423, 417406876191416533261184253, 97882032782014459518083172751
Offset: 0

Views

Author

Alois P. Heinz, Mar 10 2015

Keywords

Examples

			a(0) = 1: {}.
a(1) = 2: {{1},{1}}, {{1,1}}.
a(2) = 23: {{1},{1},{1},{2}}, {{1},{1},{2},{2}}, {{1},{2},{2},{2}}, {{1},{1},{1,2}}, {{1},{1},{2,2}}, {{1},{2},{1,1}}, {{1},{2},{1,2}}, {{1},{2},{2,2}}, {{2},{2},{1,1}}, {{2},{2},{1,2}}, {{1,1},{1,2}}, {{1,1},{2,2}}, {{1,2},{1,2}}, {{1,2},{2,2}}, {{1},{1,1,2}}, {{1},{1,2,2}}, {{1},{2,2,2}}, {{2},{1,1,1}}, {{2},{1,1,2}}, {{2},{1,2,2}}, {{1,1,1,2}}, {{1,1,2,2}}, {{1,2,2,2}}.

Links

Crossrefs

Cf. A255903.

Formula

a(n) = A255903(2n,n).

A255943 Number of collections of nonempty multisets with a total of n objects of exactly three colors.

Original entry on oeis.org

5, 33, 141, 492, 1518, 4320, 11567, 29601, 72981, 174652, 407367, 929859, 2082284, 4585752, 9947997, 21290967, 45007311, 94072878, 194580977, 398589723, 809124321, 1628620153, 3252021750, 6444843618, 12681419489, 24784331646, 48126098418, 92876836752
Offset: 3

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

Column k=3 of A255903.

Formula

a(n) = A255903(n,3).

A255944 Number of collections of nonempty multisets with a total of n objects of exactly four colors.

Original entry on oeis.org

15, 144, 848, 3936, 15800, 57420, 194034, 619912, 1894152, 5579536, 15939998, 44363268, 120707904, 321973720, 843826794, 2176829360, 5535969328, 13896678404, 34470255256, 84565452840, 205353216490, 493934650180, 1177496577964, 2783578273632, 6528396084669
Offset: 4

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

Column k=4 of A255903.

Formula

a(n) = A255903(n,4).

A255945 Number of collections of nonempty multisets with a total of n objects of exactly five colors.

Original entry on oeis.org

52, 675, 5190, 30710, 154410, 693667, 2867370, 11114495, 40923980, 144467430, 492315490, 1628069915, 5246037340, 16524577355, 51016494920, 154706348228, 461635818560, 1357490142605, 3938873183155, 11289577084470, 31993411507279, 89716807311725, 249130011081065
Offset: 5

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

Column k=5 of A255903.

Formula

a(n) = A255903(n,5).

A255946 Number of collections of nonempty multisets with a total of n objects of exactly six colors.

Original entry on oeis.org

203, 3396, 32835, 240012, 1470702, 7966632, 39371078, 181183512, 787459878, 3265329848, 13017110235, 50179023594, 187903789762, 686026805988, 2449214057754, 8571407147468, 29464524350559, 99657684426138, 332136267690149, 1092082375636980, 3546446343936000
Offset: 6

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

Column k=6 of A255903.

Formula

a(n) = A255903(n,6).

A255947 Number of collections of nonempty multisets with a total of n objects of exactly seven colors.

Original entry on oeis.org

877, 18270, 216006, 1904301, 13921922, 89193846, 517625311, 2780643510, 14036460424, 67312339598, 309183689612, 1368888648981, 5871001466715, 24489742456347, 99677744039362, 396936924677881, 1549990268577100, 5946246979402065, 22447258891006950
Offset: 7

Views

Author

Alois P. Heinz, Mar 11 2015

Keywords

Links

Crossrefs

Column k=7 of A255903.

Formula

a(n) = A255903(n,7).
Previous Showing 21-30 of 46 results. Next

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