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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A357185 Numbers k such that the k-th composition in standard order has the same length as the absolute value of its alternating sum.

Original entry on oeis.org

0, 1, 9, 12, 19, 22, 28, 34, 40, 69, 74, 84, 97, 104, 132, 135, 141, 144, 153, 177, 195, 198, 204, 216, 225, 240, 265, 271, 274, 283, 286, 292, 307, 310, 316, 321, 328, 355, 358, 364, 376, 386, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 544, 553
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2022

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   12: (1,3)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   40: (2,4)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
   97: (1,5,1)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)

Links

Crossrefs

See link for sequences related to standard compositions.
For sum equal to twice alternating sum we have A348614, counted by A262977.
For product equal to sum we have A335404, counted by A335405.
These compositions are counted by A357183.
This is the absolute value version of A357184, counted by A357183.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==Abs[ats[stc[#]]]&]

A357486 Heinz numbers of integer partitions with the same length as alternating sum.

Original entry on oeis.org

1, 2, 10, 20, 21, 42, 45, 55, 88, 91, 105, 110, 125, 156, 176, 182, 187, 198, 231, 245, 247, 312, 340, 351, 374, 390, 391, 396, 429, 494, 532, 544, 550, 551, 605, 663, 680, 702, 713, 714, 765, 780, 782, 845, 891, 910, 912, 969, 975, 1012, 1064, 1073, 1078
Offset: 1

Views

Author

Gus Wiseman, Oct 01 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    10: {1,3}
    20: {1,1,3}
    21: {2,4}
    42: {1,2,4}
    45: {2,2,3}
    55: {3,5}
    88: {1,1,1,5}
    91: {4,6}
   105: {2,3,4}
   110: {1,3,5}
   125: {3,3,3}
   156: {1,1,2,6}
   176: {1,1,1,1,5}

Crossrefs

For product instead of length we have new, counted by A004526.
The version for compositions is A357184, counted by A357182.
For absolute value we have A357486, counted by A357487.
These partitions are counted by A357189.
A000041 counts partitions, strict A000009.
A000712 up to 0's counts partitions, sum = twice alt sum, rank A349159.
A001055 counts partitions with product equal to sum, ranked by A301987.
A006330 up to 0's counts partitions, sum = twice rev-alt sum, rank A349160.
A025047 counts alternating compositions.
A357136 counts compositions by alternating sum.
Cf. A051159, A131044, A262046.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],PrimeOmega[#]==ats[Reverse[primeMS[#]]]&]

A262047 Number of ordered partitions of [n] such that at least two parts have the same size.

Original entry on oeis.org

0, 0, 2, 6, 66, 510, 4280, 46536, 542962, 7074654, 101914512, 1621871196, 28087868160, 526841965260, 10641234260358, 230278335503586, 5315641087796562, 130370690653563150, 3385534274596691456, 92801584815121975452, 2677687776095609649256
Offset: 0

Views

Author

Alois P. Heinz, Sep 09 2015

Keywords

Comments

All terms are even.

Links

Crossrefs

Cf. A000670, A032011, A262046, A261982.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1,
       add(binomial(n, k)*g(k), k=0..n-1))
    end:
b:= proc(n, i, p) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1)*binomial(n, i))))
    end:
a:= n-> g(n)-b(n$2, 0):
seq(a(n), n=0..25);
  • Mathematica
    g[n_] := g[n] = If[n<2, 1, Sum[Binomial[n, k]*g[k], {k, 0, n-1}]]; b[n_, i_, p_] := b[n, i, p] = If[i*(i+1)/2n, 0, b[n-i, i-1, p+1]*Binomial[n, i]]]]; a[n_] := g[n] - b[n, n, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

Formula

a(n) = A000670(n) - A032011(n).

A357709 Number of integer partitions of n whose length is twice their alternating sum.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 6, 6, 9, 11, 13, 18, 21, 28, 32, 44, 49, 65, 76, 96, 114, 141, 170, 204, 250, 295, 361, 425, 516, 606, 734, 858, 1031, 1210, 1440, 1690, 2000, 2347, 2759, 3240, 3786, 4441, 5174, 6053, 7030, 8210, 9509, 11074, 12807, 14870
Offset: 0

Views

Author

Gus Wiseman, Oct 16 2022

Keywords

Comments

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The alternating sum of a partition is also the number of odd conjugate parts.

Examples

			The a(1) = 0 through a(12) = 6 partitions:
  .  .  21  .  32  3111  43  3221  54      3331  65      4332
                             4211  411111  4222  422111  4431
                                           4321  521111  5322
                                           5311          5421
                                                         6411
                                                         51111111

Crossrefs

This is the "twice" version of A357189, ranked by A357486.
The version for compositions is A357847.
These partitions are ranked by A357848.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
A357182 counts compositions w/ length = alternating sum, ranked by A357184.
Cf. A004526, A262046, A262977, A301987, A357183, A357485, A357488.

Programs

  • Mathematica
    ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,30}]
Previous Showing 11-14 of 14 results.

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