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.

Previous Showing 41-49 of 49 results.

A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 9, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 19, 3, 3, 3, 24, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 46, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 37, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 46, 9, 3
Offset: 1

Views

Author

Gus Wiseman, Apr 26 2025

Keywords

Examples

			The a(36) = 24 choices:
  {{2,2,3,3}}  {{2},{2,3,3}}  {{2},{3},{2,3}}
  {{2,2,9}}    {{3},{2,2,3}}  {{2},{3},{6}}
  {{2,3,6}}    {{2,2},{3,3}}
  {{2,18}}     {{2},{2,9}}
  {{3,3,4}}    {{9},{2,2}}
  {{3,12}}     {{2},{3,6}}
  {{4,9}}      {{3},{2,6}}
  {{6,6}}      {{6},{2,3}}
  {{36}}       {{2},{18}}
               {{3},{3,4}}
               {{4},{3,3}}
               {{3},{12}}
               {{4},{9}}

Crossrefs

The case of a unique choice (positions of 1) is A008578.
This is the strict case of A050336.
For distinct strict blocks we have A050345.
For integer partitions we have A261049, strict case of A001970.
For strict blocks that are not necessarily distinct we have A296119.
Twice-partitions of this type are counted by A296122.
For normal multisets we have A317776, strict case of A255906.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, strict A296121, see A296118, A296120.
Cf. A000009, A005117, A045782, A050342, A279785, A293243, A293511, A302494, A316439, A358914, A381992, A382201.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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[Sum[Length[Select[mps[y],UnsameQ@@#&]],{y,facs[n]}],{n,30}]

A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 13, 16, 19, 22, 26, 30, 34, 38, 44, 49, 54, 62, 67, 74, 83, 89, 98, 107, 115, 124, 134, 145, 155, 168, 178, 189, 206, 217, 231, 247, 259, 277, 294, 310, 327, 345, 365, 382, 404, 424, 444, 470, 489, 513, 539, 561, 588, 613, 641, 670, 699, 729, 756, 791, 824
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

Comments

Number of partitions of n into distinct terms of A000009, where 2 different parts of 1 and 2 different parts of 2 are available (1a, 1b, 2a, 2b, 3a, 4a, 5a, 6a, ...).

Examples

			a(3) = 5 because we have [3a], [2a, 1a], [2a, 1b], [2b, 1a] and [2b, 1b].

Links

Crossrefs

Cf. A000009, A007279, A050342, A068006, A089254, A089259, A280253, A284908.

Programs

  • Maple
    N:= 20: # to get a(0) .. a(A000009(N))
P:= mul(1+x^k,k=1..N):
R:= mul(1+x^coeff(P,x,n)),n=1..N):
seq(coeff(R,x,n),n=0..coeff(P,x,N)); # Robert Israel, Sep 01 2017
  • Mathematica
    nmax = 61; CoefficientList[Series[Product[1 + x^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A000009(k)).

A304783 Expansion of Product_{k>=1} (1 - x^k)^q(k), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, -1, -1, -1, 0, 1, 0, 3, 2, 3, 1, 3, -2, 0, -6, -8, -12, -14, -18, -19, -19, -15, -3, 4, 29, 46, 90, 114, 165, 192, 248, 252, 276, 232, 185, 29, -143, -454, -811, -1324, -1909, -2609, -3348, -4132, -4851, -5386, -5653, -5380, -4470, -2477, 664, 5582, 12193, 21314
Offset: 0

Views

Author

Ilya Gutkovskiy, May 18 2018

Keywords

Comments

Convolution inverse of A089259.

Links

Crossrefs

Cf. A000009, A050342, A089254, A089259, A300508.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[Product[(1 - x^k)^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d PartitionsQ[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

Formula

G.f.: Product_{k>=1} (1 - x^k)^A000009(k).

A330456 Number of multisets of nonempty sets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 43, 84, 168, 332, 650, 1255, 2428, 4636, 8827, 16702, 31457, 58919, 109977, 204286, 378135, 697240, 1281315, 2346612, 4284654, 7799248, 14157079, 25626996, 46269838, 83330373, 149717844, 268371413, 479996794, 856661792, 1525761119, 2712050472
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2019

Keywords

Examples

			The a(4) = 10 partitions:
  ((4))  ((13))      ((1)(12))        ((2))((2))  ((1))((1))((1))((1))
         ((1)(3))    ((1))((12))
         ((1))((3))  ((1))((1)(2))
                     ((1))((1))((2))

Crossrefs

Cf. A001055, A001970, A007713, A050342, A050343, A063834, A089259, A270995, A279785, A330461, A323787-A323795, A330452-A330459.

Programs

  • Mathematica
    ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[And@@UnsameQ@@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]

Formula

Euler transform of A050342. The Euler transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} 1/(1 - x^i)^s_i.

A383311 Number of ways to choose a set multipartition (multiset of sets) of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 20, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 22, 3, 12, 1, 7, 3, 12, 1, 49, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3
Offset: 1

Views

Author

Gus Wiseman, Apr 28 2025

Keywords

Comments

First differs from A296119 at a(36) = 20, A296119(36) = 21.

Examples

			The a(36) = 20 choices are:
  {{2,3,6}}  {{2,3},{2,3}}  {{2},{3},{2,3}}  {{2},{2},{3},{3}}
  {{2,18}}   {{2},{2,9}}    {{2},{2},{9}}
  {{3,12}}   {{2},{3,6}}    {{2},{3},{6}}
  {{4,9}}    {{3},{2,6}}    {{3},{3},{4}}
  {{36}}     {{6},{2,3}}
             {{2},{18}}
             {{3},{3,4}}
             {{3},{12}}
             {{4},{9}}
             {{6},{6}}

Crossrefs

The case of a unique choice (positions of 1) is A008578.
For multisets of multisets we have A050336.
For sets of sets we have A050345.
For normal multisets we have A116540, strong A330783.
For integer partitions instead of factorizations we have A089259.
Twice-partitions of this type are counted by A270995.
For sets of multisets we have A383310 (distinct products A296118).
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, see A294788, A296120, A296121.
A302478 gives MM-numbers of set multipartitions.
A302494 gives MM-numbers of sets of sets.
Cf. A000009, A001970, A005117, A050342, A116539, A279785, A293243, A293511, A296119, A316439, A381992, A382077.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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[Sum[Length[Select[mps[y], And@@UnsameQ@@@#&]], {y,facs[n]}],{n,100}]

A225973 Number of union-closed partitions of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 12, 16, 22, 30, 39, 52, 67, 84, 112, 140, 176, 220, 282, 336, 434, 527, 660, 798, 998, 1186, 1480, 1767, 2165, 2586, 3168, 3732, 4556, 5389, 6482, 7654, 9211, 10789, 12937, 15153, 18037, 21086, 25060, 29159, 34527, 40172, 47301, 54927
Offset: 0

Views

Author

Allan C. Wechsler, May 26 2013

Keywords

Comments

The objects being counted are sets of sets of positive integers; each such set is closed under set union, and the sum of all the elements of its elements is n.
The sequence is related to Frankl's notorious union-closed sets conjecture, see the Wikipedia link.

Examples

			For n = 5, the a(5) = 5 union-closed partitions are: {{5}}, {{4,1}}, {{3,2}}, {{3,1},{1}}, {{2,1},{2}}.
{{3},{2}} has the correct sum, but is not closed under union.

References

  • This sequence was proposed by David S. Newman, in a note to the SeqFan mailing list, dated May 19 2013.

Links

Crossrefs

Cf. A050342 (answers a similar question without the requirement that the sets be closed under union).

A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 13, 19, 28, 47, 64, 98
Offset: 0

Views

Author

Gus Wiseman, Oct 12 2018

Keywords

Examples

			The a(1) = 1 through a(8) = 19 antichains:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1,3}}    {{1,4}}    {{1,5}}
                {{1},{2}}  {{1},{3}}  {{2,3}}    {{2,4}}
                                      {{1},{4}}  {{1,2,3}}
                                      {{2},{3}}  {{1},{5}}
                                                 {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{2},{1,3}}
                                                 {{3},{1,2}}
                                                 {{1},{2},{3}}
.
  {{7}}          {{8}}
  {{1,6}}        {{1,7}}
  {{2,5}}        {{2,6}}
  {{3,4}}        {{3,5}}
  {{1,2,4}}      {{1,2,5}}
  {{1},{6}}      {{1,3,4}}
  {{2},{5}}      {{1},{7}}
  {{3},{4}}      {{2},{6}}
  {{1},{2,4}}    {{3},{5}}
  {{2},{1,4}}    {{1},{2,5}}
  {{4},{1,2}}    {{1},{3,4}}
  {{1,2},{1,3}}  {{2},{1,5}}
  {{1},{2},{4}}  {{3},{1,4}}
                 {{4},{1,3}}
                 {{5},{1,2}}
                 {{1,2},{1,4}}
                 {{1,2},{2,3}}
                 {{1},{2},{5}}
                 {{1},{3},{4}}

Crossrefs

Cf. A001970, A050342, A089259, A258466, A261049, A319719, A319721, A320328, A320331, A320353.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

A330454 Number of sets of nonempty sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 2, 7, 15, 39, 94, 224, 526, 1236, 2857, 6568, 15003, 34030, 76757, 172216, 384386, 853960, 1888891, 4160524, 9128355, 19953661, 43463021, 94354292, 204182435, 440505489, 947590424, 2032730905, 4348897216, 9280361316, 19755155955, 41953293592, 88891338202
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2019

Keywords

Examples

			The a(4) = 15 partitions:
  ((4))  ((22))  ((13))      ((112))        ((1111))
                 ((1)(3))    ((1)(12))      ((1)(111))
                 ((1))((3))  ((2)(11))      ((1))((111))
                             ((1))((12))    ((1))((1)(11))
                             ((2))((11))
                             ((1))((1)(2))

Crossrefs

Cf. A001970, A007713, A050336, A050342, A050343, A261049, A271619, A279785, A330461, A330463, A323787-A323795, A330452-A330459.

Programs

  • Mathematica
    ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[UnsameQ@@#,And@@UnsameQ@@@#]&]],{n,0,10}]

Formula

Weigh transform of A261049. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

A330458 Number of multisets of nonempty sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 3, 8, 20, 49, 123, 292, 701, 1653, 3874, 8977, 20711, 47344, 107692, 243382, 547264, 1224048, 2725483, 6040796, 13334354, 29316445, 64215841, 140159357, 304890958, 661097630, 1429083295, 3080159882, 6620188725, 14190463947, 30338920339, 64702805452
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2019

Keywords

Examples

			The a(4) = 20 partitions:
  ((4))  ((22))      ((13))      ((112))          ((1111))
         ((2))((2))  ((1)(3))    ((1)(12))        ((1)(111))
                     ((1))((3))  ((2)(11))        ((1))((111))
                                 ((1))((12))      ((11))((11))
                                 ((2))((11))      ((1))((1)(11))
                                 ((1))((1)(2))    ((1))((1))((11))
                                 ((1))((1))((2))  ((1))((1))((1))((1))

Crossrefs

Cf. A001970, A007713, A050342, A050343, A063834, A089259, A261049, A270995, A271619, A323787-A323795, A330452-A330459.

Programs

  • Mathematica
    ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And@@UnsameQ@@@#&]],{n,0,10}]

Formula

Euler transform of A261049. The Euler transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} 1/(1 - x^i)^s_i.
Previous Showing 41-49 of 49 results.

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