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 11-17 of 17 results.

A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 9, 23, 62, 161, 410, 1031, 2579, 6359, 15575, 37830, 91241, 218581, 520544, 1232431, 2902644, 6802178, 15866054, 36844016, 85202436, 196251933, 450341874, 1029709478, 2346409350, 5329371142, 12066816905, 27240224766, 61317231288, 137643961196
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2019

Keywords

Comments

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

Examples

			The a(4) = 23 partitions:
  ((4))  ((22))    ((31))      ((211))        ((1111))
         ((2)(2))  ((1)(3))    ((1)(21))      ((1)(111))
                   ((1))((3))  ((2)(11))      ((11)(11))
                               ((1)(1)(2))    ((1))((111))
                               ((1))((21))    ((1)(1)(11))
                               ((2))((11))    ((1))((1)(11))
                               ((1))((1)(2))  ((1)(1)(1)(1))
                               ((2))((1)(1))  ((11))((1)(1))
                                              ((1))((1)(1)(1))

Links

Crossrefs

The not necessarily strict case is A007713.
Cf. A001055, A001970, A050336, A050343, A089259, A261049, A271619, A316980, A318566, A323787-A323795, A330452-A330459, A330461, A330463.

Programs

  • Maple
    with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) a(n):= `if`(n<2, 1, add(a(n-k)*add(b(d)
      *d*(-1)^(k/d+1), d=divisors(k)), k=1..n)/n)
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Jul 18 2021
  • 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],UnsameQ@@#&]],{n,0,10}]

Formula

Weigh transform of A001970. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

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.

A323786 Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.

Original entry on oeis.org

1, 0, 2, 3, 19, 39, 200, 615, 2849, 11174, 52377, 239269, 1191090, 6041975, 32275288, 177797719, 1017833092, 6014562272, 36717301665, 230947360981, 1495562098099, 9956230757240, 68070158777759, 477439197541792, 3432259679880648, 25267209686664449
Offset: 0

Views

Author

Gus Wiseman, Jan 28 2019

Keywords

Comments

All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(4) = 19 multiset partitions:
  {{1111}}      {{1112}}      {{1123}}      {{1234}}
  {{11}{11}}    {{1122}}      {{11}{23}}    {{12}{34}}
  {{11}}{{11}}  {{11}{12}}    {{12}{13}}    {{12}}{{34}}
                {{11}{22}}    {{11}}{{23}}
                {{12}{12}}    {{12}}{{13}}
                {{11}}{{12}}
                {{11}}{{22}}
                {{12}}{{12}}
Non-isomorphic representatives of the a(5) = 39 multiset partitions:
  {{11111}}      {{11112}}      {{11123}}      {{11234}}      {{12345}}
  {{11}{111}}    {{11122}}      {{11223}}      {{11}{234}}    {{12}{345}}
  {{11}}{{111}}  {{11}{112}}    {{11}{123}}    {{12}{134}}    {{12}}{{345}}
                 {{11}{122}}    {{11}{223}}    {{23}{114}}
                 {{12}{111}}    {{12}{113}}    {{11}}{{234}}
                 {{12}{112}}    {{12}{123}}    {{12}}{{134}}
                 {{22}{111}}    {{13}{122}}    {{23}}{{114}}
                 {{11}}{{112}}  {{23}{111}}
                 {{11}}{{122}}  {{11}}{{123}}
                 {{12}}{{111}}  {{11}}{{223}}
                 {{12}}{{112}}  {{12}}{{113}}
                 {{22}}{{111}}  {{12}}{{123}}
                                {{13}}{{122}}
                                {{23}}{{111}}

Crossrefs

Cf. A007716, A302545, A306186, A317791, A318564, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp(sExp(A-x*sv(1)))))} \\ Andrew Howroyd, Jan 17 2023

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 17 2023

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.

A330455 Number of sets of nonempty multisets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 2, 6, 12, 28, 62, 134, 285, 610, 1277, 2661, 5506, 11305, 23064, 46803, 94406, 189484, 378522, 752668, 1490319, 2939093, 5774065, 11302564, 22048496, 42869613, 83091843, 160569590, 309398958, 594532990, 1139416396, 2178119059, 4153507514, 7901706341
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2019

Keywords

Examples

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

Crossrefs

Cf. A001055, A001970, A007713, A050336, A050343, A089259, A261049, A270995, A271619, A323787-A323795, A330452-A330459, A330461.

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@@@Join@@#]&]],{n,0,10}]

Formula

Weigh transform of A089259. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

A330457 Number of multisets of nonempty multisets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 3, 7, 17, 37, 87, 187, 414, 887, 1903, 4008, 8437, 17519, 36255, 74384, 151898, 308129, 622269, 1249768, 2499392, 4975421, 9865122, 19481300, 38331536, 75149380, 146840801, 285990797, 555297342, 1074996017, 2075201544, 3995079507, 7671034324, 14692086594
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2019

Keywords

Examples

			The a(4) = 17 partitions:
  ((4))  ((13))      ((1)(12))        ((2)(2))    ((1)(1)(1)(1))
         ((1)(3))    ((1)(1)(2))      ((2))((2))  ((1))((1)(1)(1))
         ((1))((3))  ((1))((12))                  ((1)(1))((1)(1))
                     ((1))((1)(2))                ((1))((1))((1)(1))
                     ((2))((1)(1))                ((1))((1))((1))((1))
                     ((1))((1))((2))

Crossrefs

Cf. A001970, A007713, A049311, A050343, A063834, A089259, A270995, A318564, 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@@@Join@@#&]],{n,0,10}]

Formula

Euler transform of A089259. 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.

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 11-17 of 17 results.

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