A357189 -id:A357189 - cp's OEIS frontend

A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.

1, 0, 1, 2, 4, 5, 9, 13, 23, 34, 54, 78, 120, 170, 252, 358, 517, 725, 1030, 1427, 1992, 2733, 3759, 5106, 6946, 9345, 12577, 16788, 22384, 29641, 39199, 51529, 67626, 88307, 115083, 149332, 193383, 249456, 321134, 411998, 527472, 673233, 857539, 1089223, 1380772

Offset: 1

Gus Wiseman, Oct 02 2022

nonn

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(7) = 9 partitions:
  (1)  .  (311)  (322)  (333)    (443)    (553)
                 (421)  (432)    (542)    (652)
                        (531)    (641)    (751)
                        (51111)  (52211)  (52222)
                                 (62111)  (53311)
                                          (62221)
                                          (63211)
                                          (73111)
                                          (7111111)

For product equal to sum we have A001055, compositions A335405.
The version for compositions appears to be A222763, odd version of A357182.
These are the odd-indexed terms of A357189, ranked by A357486.
These partitions are ranked by the odd-sum portion of A357485.
Except at the start, alternately adding zeros gives A357487.
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.
Cf. A004526, A051159, A114220, A131044, A262046, A262977, A357183, A357184.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,1,30,2}]

a(n) = A357189(2n - 1).

More terms from Alois P. Heinz, Oct 04 2022

A357632 Numbers k such that the skew-alternating sum of the prime indices of k is 0.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 90, 100, 121, 144, 169, 196, 210, 225, 256, 289, 324, 360, 361, 400, 441, 462, 484, 525, 529, 550, 576, 625, 676, 729, 784, 840, 841, 858, 900, 910, 961, 1024, 1089, 1155, 1156, 1225, 1296, 1326, 1369, 1440, 1444, 1521, 1600

Offset: 1

Gus Wiseman, Oct 09 2022

nonn

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 + ....

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    36: {1,1,2,2}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    90: {1,2,2,3}
   100: {1,1,3,3}
   121: {5,5}
   144: {1,1,1,1,2,2}

The version for original alternating sum is A000290.
The version for standard compositions is A357627, reverse A357628.
Positions of zeros in A357630, reverse A357634.
The half-alternating form is A357631, reverse A357635.
The reverse version is A357636.
These partitions are counted by A357640, half A357639.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[1000],skats[primeMS[#]]==0&]

A357633 Half-alternating sum of the partition having Heinz number n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 1, 4, 4, 5, 2, 6, 5, 5, 0, 7, 3, 8, 3, 6, 6, 9, 1, 6, 7, 2, 4, 10, 4, 11, 1, 7, 8, 7, 2, 12, 9, 8, 2, 13, 5, 14, 5, 3, 10, 15, 2, 8, 5, 9, 6, 16, 1, 8, 3, 10, 11, 17, 3, 18, 12, 4, 2, 9, 6, 19, 7, 11, 6, 20, 3, 21, 13, 4, 8, 9, 7, 22, 3, 0

Offset: 1

Gus Wiseman, Oct 09 2022

nonn

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

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 partition with Heinz number 525 is (4,3,3,2) so a(525) = 4 + 3 - 3 - 2 = 2.

The original alternating sum is A316524, reverse A344616.
The version for standard compositions is A357622, non-reverse A357621.
The skew-alternating form is A357634, non-reverse A357630.
Positions of zeros are A000583, non-reverse A357631.
The reverse version is A357629.
These partitions are counted by A357637, skew A357638.
A056239 adds up prime indices, row sums of A112798.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357623-A357626, A357632, A357636, A357640.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[halfats[Reverse[primeMS[n]]],{n,30}]

A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 2, 5, 12, 22, 26, 58, 100, 203, 282, 616, 962, 2045, 2982, 6518, 9858, 21416, 31680, 69623, 104158, 228930, 339978, 751430, 1119668, 2478787, 3684082, 8182469, 12171900, 27082870, 40247978, 89748642, 133394708, 297933185, 442628598, 990210110

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (13)  (113)  (24)  (124)  (35)
       (31)  (212)  (42)  (151)  (53)
             (311)        (223)  (1115)
                          (322)  (1151)
                          (421)  (1214)
                                 (1313)
                                 (1412)
                                 (1511)
                                 (2141)
                                 (3131)
                                 (4121)
                                 (5111)

For product instead of length we have A114220.
For sum equal to twice alternating sum we have A262977, ranked by A348614.
For product equal to sum we have A335405, ranked by A335404.
This is the absolute value version of A357182.
These compositions are ranked by A357185.
The case of partitions is A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A261983 counts non-anti-run compositions.
A357136 counts compositions by alternating sum.
Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==Abs[ats[#]]&]],{n,0,15}]

a(21)-a(39) from Alois P. Heinz, Sep 29 2022

A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 49, 63, 64, 70, 81, 90, 100, 108, 121, 144, 154, 165, 169, 192, 196, 210, 225, 256, 273, 286, 289, 300, 324, 325, 360, 361, 400, 441, 442, 462, 480, 484, 525, 529, 550, 561, 576, 588, 595, 625, 646, 676, 700, 729, 741, 750, 784

Offset: 1

Gus Wiseman, Oct 09 2022

nonn

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 + ....

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 terms together with their prime indices begin:
    1: {}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   25: {3,3}
   30: {1,2,3}
   36: {1,1,2,2}
   49: {4,4}
   63: {2,2,4}
   64: {1,1,1,1,1,1}
   70: {1,3,4}
   81: {2,2,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}

The version for original alternating sum is A000290.
The half-alternating form is A000583, non-reverse A357631.
The version for standard compositions is A357628, non-reverse A357627.
The non-reverse version is A357632.
Positions of zeros in A357634, non-reverse A357630.
These partitions are counted by A357640, half A357639.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even A357642.
Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[1000],skats[Reverse[primeMS[#]]]==0&]

A035544 Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 4, 0, 10, 0, 13, 0, 28, 0, 37, 0, 72, 0, 96, 0, 172, 0, 230, 0, 391, 0, 522, 0, 846, 0, 1129, 0, 1766, 0, 2348, 0, 3564, 0, 4722, 0, 6992, 0, 9226, 0, 13371, 0, 17568, 0, 25006, 0, 32708, 0, 45817, 0, 59668, 0, 82430, 0, 106874, 0, 145830, 0, 188260, 0

Offset: 0

Olivier Gérard

nonn

From Gus Wiseman, Oct 12 2022: (Start)
Also the number of integer partitions of n whose skew-alternating sum is 0, where 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 + ... These are the conjugates of the partitions described in the name. For example, the a(0) = 1 through a(8) = 10 partitions are:
  ()  .  (11)  .  (22)    .  (33)      .  (44)
                  (211)      (321)        (422)
                  (1111)     (2211)       (431)
                             (111111)     (2222)
                                          (3221)
                                          (3311)
                                          (22211)
                                          (221111)
                                          (2111111)
                                          (11111111)
The ordered version (compositions) is A138364
These partitions are ranked by A357636, reverse A357632.
The reverse version is A357640 (aerated).
Cf. A357623, A357629, A357630, A357634, A357646, A357705.
(End)

			From _Gus Wiseman_, Oct 12 2022: (Start)
The a(0) = 1 through a(8) = 10 partitions:
  ()  .  (2)  .  (4)   .  (6)    .  (8)
                 (22)     (42)      (44)
                 (31)     (222)     (53)
                          (321)     (62)
                                    (71)
                                    (422)
                                    (431)
                                    (2222)
                                    (3221)
                                    (3311)
(End)

The case with at least one odd part is A035550.
Removing zeros gives A035594.
Central column k=0 of A357638.
These partitions are ranked by A357707.
A000041 counts integer partitions.
A344651 counts partitions by alternating sum, ordered A097805.
Cf. A035363, A053251, A298311, A357189, A357487, A357488.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[n],skats[#]==0&]],{n,0,30}] (* Gus Wiseman,Oct 12 2022 *)

More terms from David W. Wilson

A357485 Heinz numbers of integer partitions with the same length as reverse-alternating sum.

Original entry on oeis.org

1, 2, 20, 42, 45, 105, 110, 125, 176, 182, 231, 245, 312, 374, 396, 429, 494, 605, 663, 680, 702, 780, 782, 845, 891, 969, 1064, 1088, 1100, 1102, 1311, 1426, 1428, 1445, 1530, 1755, 1805, 1820, 1824, 1950, 2001, 2024, 2146, 2156, 2394, 2448, 2475, 2508, 2542

Offset: 1

Gus Wiseman, Oct 01 2022

nonn

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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    20: {1,1,3}
    42: {1,2,4}
    45: {2,2,3}
   105: {2,3,4}
   110: {1,3,5}
   125: {3,3,3}
   176: {1,1,1,1,5}
   182: {1,4,6}
   231: {2,4,5}
   245: {3,4,4}
   312: {1,1,1,2,6}
   374: {1,5,7}
   396: {1,1,2,2,5}

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

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[primeMS[#]]&]

A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 2, 0, 4, 0, 5, 0, 9, 0, 13, 0, 23, 0, 34, 0, 54, 0, 78, 0, 120, 0, 170, 0, 252, 0, 358, 0, 517, 0, 725, 0, 1030, 0, 1427, 0, 1992, 0, 2733, 0, 3759, 0, 5106, 0, 6946, 0, 9345, 0, 12577, 0, 16788, 0, 22384, 0, 29641, 0

Offset: 0

Gus Wiseman, Oct 01 2022

nonn

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

			The a(1) = 1 through a(13) = 9 partitions:
  1   .  .  .  311   .  322   .  333     .  443     .  553
                        421      432        542        652
                                 531        641        751
                                 51111      52211      52222
                                            62111      53311
                                                       62221
                                                       63211
                                                       73111
                                                       7111111

For product equal to sum we have A001055, compositions A335405.
The version for compositions is A357182, reverse ranked by A357184.
The reverse version is A357189, ranked by A357486.
These partitions are ranked by A357485.
Removing zeros gives A357488.
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.
Cf. A004526, A051159, A114220, A131044, A262046, A262977, A301987, A357183.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[Reverse[#]]&]],{n,0,30}]

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

Gus Wiseman, Oct 01 2022

nonn

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.

			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}

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.

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[#]]]&]

A357704 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 3, 0, 0, 2, 2, 0, 3, 0, 0, 3, 1, 3, 0, 4, 0, 0, 3, 2, 4, 2, 0, 4, 0, 0, 4, 2, 6, 2, 3, 0, 5, 0, 0, 4, 3, 5, 7, 3, 3, 0, 5, 0, 0, 5, 3, 8, 4, 10, 2, 4, 0, 6, 0, 0, 5, 4, 8, 6, 11, 9, 3, 4, 0, 6, 0, 0, 6, 4, 11, 5, 15, 8, 13, 3, 5, 0, 7

Offset: 0

Gus Wiseman, Oct 10 2022

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

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  2  0  3
  0  0  2  2  0  3
  0  0  3  1  3  0  4
  0  0  3  2  4  2  0  4
  0  0  4  2  6  2  3  0  5
  0  0  4  3  5  7  3  3  0  5
  0  0  5  3  8  4 10  2  4  0  6
  0  0  5  4  8  6 11  9  3  4  0  6
  0  0  6  4 11  5 15  8 13  3  5  0  7
  0  0  6  5 11  8 13 19 10 13  4  5  0  7
  0  0  7  5 14  8 19 13 25  9 17  4  6  0  8
  0  0  7  6 14 11 19 17 29 23 13 18  5  6  0  8
Row n = 7 counts the following reversed partitions:
  .  .  (115)   (124)   (133)      (11113)   .  (7)
        (1114)  (1222)  (223)      (111112)     (16)
        (1123)          (11122)                 (25)
                        (1111111)               (34)

Row sums are A000041.
Last entry of row n is A008619(n).
The central column in the non-reverse case is A035363, skew A035544.
For original reverse-alternating sum we have A344612.
For original alternating sum we have A344651, ordered A097805.
The non-reverse version is A357637, skew A357638.
The central column is A357639, skew A357640.
The non-reverse ordered version (compositions) is A357645, skew A357646.
The skew-alternating version is A357705.
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. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],halfats[#]==k&]],{n,0,15},{k,-n,n,2}]

cp's OEIS Frontend

A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.

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A357632 Numbers k such that the skew-alternating sum of the prime indices of k is 0.

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A357633 Half-alternating sum of the partition having Heinz number n.

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A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

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A357636 Numbers k such that the skew-alternating sum of the partition having Heinz number k is 0.

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A035544 Number of partitions of n with equal number of parts congruent to each of 1 and 3 (mod 4).

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A357485 Heinz numbers of integer partitions with the same length as reverse-alternating sum.

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A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

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

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