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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A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 3, 2, 3, 2, 0, 1, 0, 4, 2, 4, 1, 3, 0, 1, 0, 4, 3, 3, 6, 2, 3, 0, 1, 0, 5, 3, 5, 3, 7, 2, 4, 0, 1, 0, 5, 4, 5, 4, 9, 7, 3, 4, 0, 1, 0, 6, 4, 7, 3, 12, 5, 10, 3, 5, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Oct 10 2022

Keywords

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  3  1  2  0  1
  0  3  2  3  2  0  1
  0  4  2  4  1  3  0  1
  0  4  3  3  6  2  3  0  1
  0  5  3  5  3  7  2  4  0  1
  0  5  4  5  4  9  7  3  4  0  1
  0  6  4  7  3 12  5 10  3  5  0  1
  0  6  5  7  5 10 16  7 11  4  5  0  1
  0  7  5  9  5 14 11 18  7 14  4  6  0  1
Row n = 7 counts the following reversed partitions:
  .  (16)   (25)   (34)       (1123)  (1114)   .  (7)
     (115)  (223)  (1222)             (11113)
     (124)         (111112)           (11122)
     (133)         (1111111)

Crossrefs

Row sums are A000041.
First nonzero entry of each row is A004526.
The central column is A357640, half A357639.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357704.
The ordered non-reverse version (compositions) is A357646, half A357645.
The non-reverse version is A357638, half A357637.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew A357634.
Cf. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.

Programs

  • Mathematica
    skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],skats[#]==k&]],{n,0,11},{k,-n,n,2}]

A357847 Number of integer compositions of n whose length is twice their alternating sum.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 3, 1, 8, 11, 15, 46, 59, 127, 259, 407, 888, 1591, 2925, 5896, 10607, 20582, 39446, 73448, 142691, 269777, 513721, 988638, 1876107, 3600313, 6893509, 13165219, 25288200, 48408011, 92824505, 178248758, 341801149, 656641084, 1261298356
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.

Examples

			The a(0) = 1 through a(9) = 15 compositions:
  ()  .  .  (21)  .  (32)  (1131)  (43)  (1142)  (54)
                           (2121)        (1241)  (111141)
                           (3111)        (2132)  (112131)
                                         (2231)  (113121)
                                         (3122)  (114111)
                                         (3221)  (211131)
                                         (4112)  (212121)
                                         (4211)  (213111)
                                                 (311121)
                                                 (312111)
                                                 (411111)

Links

Crossrefs

The version for partitions is A357709, ranked by A357848.
A011782 counts compositions.
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.
A357189 counts partitions w/ length = alternating sum, ranked by A357486.
Cf. A262977, A301987, A357183, A357485, A357488.

Programs

  • Mathematica
    ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,10}]

Extensions

a(21)-a(38) from Alois P. Heinz, Oct 19 2022

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}]

A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.

Original entry on oeis.org

1, 6, 15, 35, 40, 77, 84, 90, 143, 189, 210, 220, 221, 224, 250, 323, 364, 437, 462, 490, 495, 504, 525, 528, 667, 748, 819, 858, 899, 988, 1029, 1040, 1134, 1147, 1155, 1188, 1210, 1320, 1326, 1375, 1400, 1408, 1517, 1564, 1683, 1690, 1763, 1904, 1938, 2021
Offset: 1

Views

Author

Gus Wiseman, Oct 16 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: {}
     6: {1,2}
    15: {2,3}
    35: {3,4}
    40: {1,1,1,3}
    77: {4,5}
    84: {1,1,2,4}
    90: {1,2,2,3}
   143: {5,6}
   189: {2,2,2,4}
   210: {1,2,3,4}
   220: {1,1,3,5}
   221: {6,7}
   224: {1,1,1,1,1,4}

Crossrefs

These partitions are counted by A357709.
The version for compositions is counted by A357847.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A025047 counts alternating compositions.
A056239 adds up prime indices.
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.
A357189 counts partitions w/ length = alternating sum, ranked by A357486.
Cf. A000720, A001221, A001222, A262977, A301987, A357183, A357485, A357488.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[1000],Length[primeMS[#]]==2sats[primeMS[#]]&]
Previous Showing 21-24 of 24 results.

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