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 11-16 of 16 results.

A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 68, 100, 153, 227, 342, 509, 759, 1129, 1678, 2492, 3699, 5477, 8121, 12015, 17795, 26313, 38924, 57541, 85065, 125712, 185758, 274431, 405420, 598815, 884465, 1306165, 1928943, 2848360, 4205979, 6210289, 9169540
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2022

Keywords

Examples

			The a(0) = 1 through a(6) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (33)
                 (111)  (31)    (32)     (42)
                        (211)   (41)     (51)
                        (1111)  (221)    (222)
                                (311)    (231)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Links

Crossrefs

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358901.
The version not counting multiplicity is A358902, strict A358903.
The case of partitions is A358909, complement A358910.
The case of equality is A358911, partitions A319169.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.

Programs

  • Mathematica
    Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,10}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 84, 134, 213, 338, 536, 850, 1349, 2136, 3389, 5367, 8509, 13480, 21362, 33843, 53624, 84957, 134600, 213251, 337850, 535251, 847987, 1343440, 2128372, 3371895, 5341977, 8463051, 13407689, 21241181, 33651507, 53312538, 84460690
Offset: 0

Views

Author

Gus Wiseman, Dec 07 2022

Keywords

Examples

			The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (24)
                 (111)  (31)    (32)     (33)
                        (211)   (41)     (42)
                        (1111)  (221)    (51)
                                (311)    (222)
                                (2111)   (231)
                                (11111)  (321)
                                         (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Links

Crossrefs

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358903.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A116608 counts partitions by sum and number of distinct parts.
A334028 counts distinct parts in standard compositions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A056239, A071625, A129519, A300335, A358831, A358908, A358911.

Programs

  • Maple
    p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t<=i, b(n-j, t), 0))(p(j)), j=1..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 14 2024
  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeNu/@#&]],{n,0,10}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306
Offset: 0

Views

Author

Gus Wiseman, Dec 09 2022

Keywords

Comments

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

Crossrefs

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2022

Keywords

Examples

			The a(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {2}     {3}        {4}           {5}              {6}
       {11}    {12}       {13}          {14}             {15}
       {1}{1}  {111}      {22}          {23}             {24}
               {1}{1}{1}  {112}         {113}            {33}
                          {1111}        {122}            {114}
                          {2}{2}        {1112}           {123}
                          {11}{11}      {11111}          {222}
                          {1}{1}{1}{1}  {1}{1}{1}{1}{1}  {1113}
                                                         {1122}
                                                         {3}{3}
                                                         {11112}
                                                         {111111}
                                                         {12}{12}
                                                         {2}{2}{2}
                                                         {111}{111}
                                                         {11}{11}{11}
                                                         {1}{1}{1}{1}{1}{1}

Links

Crossrefs

For just constant sums we have A305551, ranked by A326534.
For just constant lengths we have A319066, ranked by A320324.
The version for set partitions is A327899.
For distinct instead of constant lengths and sums we have A358832.
The version for twice-partitions is A358833.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Cf. A000219, A007425, A141199, A327908, A356932, A358831.

Programs

  • Mathematica
    Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]
  • PARI
    P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022

Formula

a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022

A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605
Offset: 0

Views

Author

Gus Wiseman, Dec 09 2022

Keywords

Examples

			The a(9) = 1 through a(14) = 11 partitions:
  (54)  (541)  (74)    (543)    (76)      (554)
               (542)   (741)    (544)     (743)
               (5411)  (5421)   (742)     (761)
                       (54111)  (5422)    (5432)
                                (5431)    (5441)
                                (7411)    (7421)
                                (54211)   (54221)
                                (541111)  (54311)
                                          (74111)
                                          (542111)
                                          (5411111)

Crossrefs

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The complement is counted by A358909.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358902, A358903, A358908.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A358837 Number of odd-length multiset partitions of integer partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 7, 14, 28, 54, 106, 208, 399, 757, 1424, 2642, 4860, 8851, 15991, 28673, 51095, 90454, 159306, 279067, 486598, 844514, 1459625, 2512227, 4307409, 7357347, 12522304, 21238683, 35903463, 60497684, 101625958, 170202949, 284238857, 473356564, 786196353
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2022

Keywords

Examples

			The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}    {{3}}          {{4}}            {{5}}
         {{1,1}}  {{1,2}}        {{1,3}}          {{1,4}}
                  {{1,1,1}}      {{2,2}}          {{2,3}}
                  {{1},{1},{1}}  {{1,1,2}}        {{1,1,3}}
                                 {{1,1,1,1}}      {{1,2,2}}
                                 {{1},{1},{2}}    {{1,1,1,2}}
                                 {{1},{1},{1,1}}  {{1,1,1,1,1}}
                                                  {{1},{1},{3}}
                                                  {{1},{2},{2}}
                                                  {{1},{1},{1,2}}
                                                  {{1},{2},{1,1}}
                                                  {{1},{1},{1,1,1}}
                                                  {{1},{1,1},{1,1}}
                                                  {{1},{1},{1},{1},{1}}

Links

Crossrefs

The version for set partitions is A024429.
These multiset partitions are ranked by A026424.
The version for partitions is A027193.
The version for twice-partitions is A358824.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Cf. A000219, A141199, A336342, A358334, A358831.

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[Join@@mps/@Reverse/@IntegerPartitions[n],OddQ[Length[#]]&]],{n,0,10}]
  • PARI
    P(v,y) = {1/prod(k=1, #v, (1 - y*x^k + O(x*x^#v))^v[k])}
seq(n) = {my(v=vector(n, k, numbpart(k))); (Vec(P(v,1)) - Vec(P(v,-1)))/2} \\ Andrew Howroyd, Dec 31 2022

Formula

G.f.: ((1/Product_{k>=1} (1-x^k)^A000041(k)) - (1/Product_{k>=1} (1+x^k)^A000041(k))) / 2. - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022
Previous Showing 11-16 of 16 results.

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