A357136 -id:A357136 - cp's OEIS frontend

A357641 Number of integer compositions of 2n whose half-alternating sum is 0.

1, 0, 2, 8, 28, 104, 396, 1504, 5720, 21872, 83980, 323344, 1248072, 4828784, 18721080, 72711552, 282861360, 1101980000, 4298748300, 16789002736, 65641204200, 256895795312, 1006308200040, 3945185586368, 15478849767888, 60774329914144, 238775589937976

Offset: 0

Gus Wiseman, Oct 12 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 a(0) = 1 through a(3) = 8 compositions:
  ()  .  (112)   (123)
         (1111)  (213)
                 (1212)
                 (1221)
                 (2112)
                 (2121)
                 (11121)
                 (11211)

Alois P. Heinz, Table of n, a(n) for n = 0..1664

The skew-alternating version appears to be A001700.
The version for partitions is A035363.
The skew-alternating form is A088218 (also for full alternating sum).
These compositions are ranked by A357625, reverse A357626.
For reversed partitions we have A357639, ranked by A357631.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357621 = half-alternating sum of standard compositions, reverse A357622.
A357637 counts partitions by half-alternating sum, skew A357638.
Cf. A000583, A001511, A053251, A344619, A357136, A357182, A357627, A357628, A357629, A357633, A357642.

a:= proc(n) option remember; `if`(n<3, [1, 0, 2][n+1],
      (8*(n-3)*(5*n-7)*(2*n-5)*a(n-3) -4*(5*n-12)*(n-2)^2*a(n-2)
       +2*(2*n-5)*(5*n-7)*n*a(n-1))/((5*n-12)*(n+1)*(n-2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 19 2022

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n],halfats[#]==0&]],{n,0,7}]

a(11)-a(26) from Alois P. Heinz, Oct 19 2022

A357621 Half-alternating sum of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 1, 4, 4, 4, 2, 4, 2, 0, 0, 5, 5, 5, 3, 5, 3, 1, 1, 5, 3, 1, 1, -1, -1, -1, 1, 6, 6, 6, 4, 6, 4, 2, 2, 6, 4, 2, 2, 0, 0, 0, 2, 6, 4, 2, 2, 0, 0, 0, 2, -2, -2, -2, 0, -2, 0, 2, 2, 7, 7, 7, 5, 7, 5, 3, 3, 7, 5, 3, 3, 1, 1, 1, 3, 7, 5, 3, 3, 1

Offset: 0

Gus Wiseman, Oct 07 2022

sign

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 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 358-th composition is (2,1,3,1,2) so a(358) = 2 + 1 - 3 - 1 + 2 = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The reverse version is A357622.
The skew-alternating form is A357623, reverse A357624.
Positions of zeros are A357625, reverse A357626.
The version for prime indices is A357629.
The version for Heinz numbers of partitions is A357633.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185.
Cf. A357627, A357628, A357630, A357631, A357634, A357635, A357640.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[halfats[stc[n]],{n,0,100}]

Positions of first appearances are powers of 2 and even powers of 2 times 7, or A029746 without 7.

A357637 Triangle read by rows where T(n,k) is the number of 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, 1, 1, 3, 0, 0, 0, 2, 2, 3, 0, 0, 0, 0, 5, 2, 4, 0, 0, 0, 0, 2, 6, 3, 4, 0, 0, 0, 0, 2, 3, 9, 3, 5, 0, 0, 0, 0, 0, 4, 7, 10, 4, 5, 0, 0, 0, 0, 0, 0, 11, 8, 13, 4, 6, 0, 0, 0, 0, 0, 0, 4, 15, 12, 14, 5, 6, 0, 0, 0, 0, 0, 0, 3, 7, 25, 13, 17, 5, 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  1  1  3
  0  0  0  2  2  3
  0  0  0  0  5  2  4
  0  0  0  0  2  6  3  4
  0  0  0  0  2  3  9  3  5
  0  0  0  0  0  4  7 10  4  5
  0  0  0  0  0  0 11  8 13  4  6
  0  0  0  0  0  0  4 15 12 14  5  6
  0  0  0  0  0  0  3  7 25 13 17  5  7
Row n = 9 counts the following partitions:
  (3222)       (333)      (432)     (441)  (9)
  (22221)      (3321)     (522)     (531)  (54)
  (21111111)   (4221)     (4311)    (621)  (63)
  (111111111)  (32211)    (5211)    (711)  (72)
               (222111)   (6111)           (81)
               (2211111)  (33111)
               (3111111)  (42111)
                          (51111)
                          (321111)
                          (411111)

Alois P. Heinz, Rows n = 0..200, flattened

Row sums are A000041.
Number of nonzero entries in row n appears to be A004525(n+1).
Last entry of row n is A008619(n).
Column sums appear to be A029862.
The central column is A035363, skew A035544.
For original alternating sum we have A344651, ordered A097805.
The skew-alternating version is A357638.
The central column of the reverse is A357639, skew A357640.
The ordered version (compositions) is A357645, skew A357646.
The reverse version is A357704, skew 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. A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

b:= proc(n, i, s, t) option remember; `if`(n=0, x^s, `if`(i<1, 0,
      b(n, i-1, s, t)+b(n-i, min(n-i, i), s+`if`(t<2, i, -i), irem(t+1, 4))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=-n..n, 2))(b(n$2, 0$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Oct 12 2022

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Table[Length[Select[IntegerPartitions[n],halfats[#]==k&]],{n,0,12},{k,-n,n,2}]

Conjecture: The column sums are A029862.

A357182 Number of integer compositions of n with the same length as their alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 1, 4, 6, 20, 13, 48, 50, 175, 141, 512, 481, 1719, 1491, 5400, 4929, 17776, 15840, 57420, 52079, 188656, 169989, 617176, 559834, 2033175, 1842041, 6697744, 6085950, 22139780, 20123989, 73262232, 66697354, 242931321, 221314299, 806516560

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) = 6 compositions:
  (1)  (31)  (113)  (42)  (124)  (53)
             (212)        (223)  (1151)
             (311)        (322)  (2141)
                          (421)  (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.
For absolute value we have A357183.
These compositions are ranked by A357184.
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[#]==ats[#]&]],{n,0,15}]

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

A357189 Number of integer partitions of n with the same length as alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 2, 4, 3, 5, 6, 9, 9, 13, 16, 23, 23, 34, 37, 54, 54, 78, 84, 120, 121, 170, 182, 252, 260, 358, 379, 517, 535, 725, 764, 1030, 1064, 1427, 1494, 1992, 2059, 2733, 2848, 3759, 3887, 5106, 5311, 6946, 7177, 9345, 9701, 12577, 12996, 16788

Offset: 0

Gus Wiseman, Sep 30 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(4) = 1 through a(13) = 9 partitions:
  31   311   42   322   53     333     64     443     75       553
                  421   5111   432     5221   542     5331     652
                               531     6211   641     6222     751
                               51111          52211   6321     52222
                                              62111   7311     53311
                                                      711111   62221
                                                               63211
                                                               73111
                                                               7111111

For product equal to sum we have A001055, compositions A335405.
For product instead of length we have A004526, compositions A114220.
The version for compositions is A357182, ranked by A357184.
For sum equal to twice alternating sum we have A357189 (this sequence).
These partitions are ranked by A357486.
The reverse version is A357487, ranked by A357485.
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. A051159, A070939, A131044, A262046, A262977, A301987, A357183.

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

A357638 Triangle read by rows where T(n,k) is the number of 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, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 4, 5, 4, 1, 1, 0, 0, 0, 1, 10, 5, 4, 1, 1, 0, 0, 0, 1, 5, 13, 5, 4, 1, 1, 0, 0, 0, 0, 4, 13, 14, 5, 4, 1, 1, 0, 0, 0, 0, 1, 13, 17, 14, 5, 4, 1, 1

Offset: 0

Gus Wiseman, Oct 10 2022

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 + ....

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  0  3  1  1
  0  0  1  4  1  1
  0  0  1  4  4  1  1
  0  0  0  4  5  4  1  1
  0  0  0  1 10  5  4  1  1
  0  0  0  1  5 13  5  4  1  1
  0  0  0  0  4 13 14  5  4  1  1
  0  0  0  0  1 13 17 14  5  4  1  1
  0  0  0  0  1  5 28 18 14  5  4  1  1
Row n = 7 counts the following partitions:
  .  .  .  (322)      (43)      (52)     (61)  (7)
           (331)      (421)     (511)
           (2221)     (3211)    (4111)
           (1111111)  (22111)   (31111)
                      (211111)

Row sums are A000041.
Number of nonzero entries in row n appears to be A004396(n+1).
First nonzero entry of each row appears to converge to A146325.
The central column is A035544, half A035363.
Column sums appear to be A298311.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357637.
The ordered version (compositions) is A357646, half A357645.
The reverse version is A357705, half A357704.
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. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.

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

Conjecture: The columns are palindromes with sums A298311.

A357623 Skew-alternating sum of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, -1, -1, 4, 2, 0, 0, -2, -2, -2, 0, 5, 3, 1, 1, -1, -1, -1, 1, -3, -3, -3, -1, -3, -1, 1, 1, 6, 4, 2, 2, 0, 0, 0, 2, -2, -2, -2, 0, -2, 0, 2, 2, -4, -4, -4, -2, -4, -2, 0, 0, -4, -2, 0, 0, 2, 2, 2, 0, 7, 5, 3, 3, 1, 1, 1, 3, -1, -1, -1, 1, -1

Offset: 0

Gus Wiseman, Oct 08 2022

sign

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 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 358-th composition is (2,1,3,1,2) so a(358) = 2 - 1 - 3 + 1 + 2 = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Positions of positive firsts appear to be A029744.
The half-alternating form is A357621, reverse A357622.
The reverse version is A357624.
Positions of zeros are A357627, reverse A357628.
The version for prime indices is A357630.
The version for Heinz numbers of partitions is A357634.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001700, A001511, A053251, A344619, A357136, A357182, A357183, A357184, A357185, A357625, A357626.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[skats[stc[n]],{n,0,100}]

A357625 Numbers k such that the k-th composition in standard order has half-alternating sum 0.

Original entry on oeis.org

0, 14, 15, 44, 45, 46, 52, 53, 54, 59, 61, 152, 153, 154, 156, 168, 169, 170, 172, 179, 181, 185, 200, 201, 202, 204, 211, 213, 217, 230, 231, 234, 235, 239, 242, 243, 247, 254, 255, 560, 561, 562, 564, 568, 592, 593, 594, 596, 600, 611, 613, 617, 625, 656

Offset: 1

Gus Wiseman, Oct 08 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 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 sequence together with the corresponding compositions begins:
    0: ()
   14: (1,1,2)
   15: (1,1,1,1)
   44: (2,1,3)
   45: (2,1,2,1)
   46: (2,1,1,2)
   52: (1,2,3)
   53: (1,2,2,1)
   54: (1,2,1,2)
   59: (1,1,2,1,1)
   61: (1,1,1,2,1)

John Tyler Rascoe, Table of n, a(n) for n = 1..7762
Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The version for full alternating sum is A344619.
Positions of zeros in A357621.
The reverse version is A357626.
The skew-alternating form is A357627, reverse A357628.
The version for prime indices is A357631.
The version for Heinz numbers of partitions is A357635.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357622, A357623, A357629, A357633.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Select[Range[0,100],halfats[stc[#]]==0&]

from itertools import count, islice
def comp(n): #row n of A066099 after Franklin T. Adams-Watters
    v,k = [],0
    while n > 0:
        k += 1
        if n%2 == 1:
            v.append(k)
            k = 0
        n = n//2
    return(v[::-1])
def a_gen():
    for n in count(0):
        c = comp(n)
        x = sum(c[i]*(-1)**(i//2) for i in range(len(c)))
        if x == 0:
            yield(n)
A357625_list = list(islice(a_gen(), 60)) # John Tyler Rascoe, Jun 01 2024

A357640 Number of reversed integer partitions of 2n whose skew-alternating sum is 0.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 24, 40, 59, 93, 136, 208, 299, 445, 632, 921, 1292, 1848, 2563, 3610, 4954, 6881, 9353, 12835, 17290, 23469, 31357, 42150, 55889, 74463, 98038, 129573, 169476, 222339, 289029, 376618, 486773, 630313, 810285, 1043123, 1334174

Offset: 0

Gus Wiseman, Oct 11 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 a(0) = 1 through a(5) = 9 partitions:
  ()  (11)  (22)    (33)      (44)        (55)
            (1111)  (2211)    (2222)      (3322)
                    (111111)  (3221)      (4321)
                              (3311)      (4411)
                              (221111)    (222211)
                              (11111111)  (322111)
                                          (331111)
                                          (22111111)
                                          (1111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from Lucas A. Brown)
Lucas A. Brown, A357640.py.

The non-reverse half-alternating version is A035363/A035444.
The non-reverse version appears to be A035544/A035594.
These partitions are ranked by A357632, half A357631.
The half-alternating version is A357639.
A000041 counts integer partitions (also reversed integer partitions).
A316524 gives alternating sum of prime indices, reverse A344616.
A344651 counts alternating sum of partitions by length, ordered A097805.
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.
A357637 counts partitions by half-alternating sum, skew A357638.
Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357636, A357641, A357645, A357704.

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

a(31) onwards from Lucas A. Brown, Oct 19 2022

A357642 Number of even-length integer compositions of 2n whose half-alternating sum is 0.

Original entry on oeis.org

1, 0, 1, 4, 13, 48, 186, 712, 2717, 10432, 40222, 155384, 601426, 2332640, 9063380, 35269392, 137438685, 536257280, 2094786870, 8191506136, 32063203590, 125613386912, 492516592620, 1932569186288, 7588478653938, 29816630378368, 117226929901676, 461151757861552

Offset: 0

Gus Wiseman, Oct 12 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 a(0) = 1 through a(4) = 13 compositions:
  ()  .  (1111)  (1212)  (1313)
                 (1221)  (1322)
                 (2112)  (1331)
                 (2121)  (2213)
                         (2222)
                         (2231)
                         (3113)
                         (3122)
                         (3131)
                         (111311)
                         (112211)
                         (113111)
                         (11111111)

David A. Corneth, Table of n, a(n) for n = 0..1665

The skew-alternating version appears to be A000984.
For original alternating sum we have A001700/A088218.
The version for partitions of any length is A357639, ranked by A357631.
For length multiple of 4 we have A110145.
These compositions of any length are ranked by A357625, reverse A357626.
A124754 gives alternating sum of standard compositions, reverse A344618.
A357621 = half-alternating sum of standard compositions, reverse A357622.
A357637 counts partitions by half-alternating sum, skew A357638.
Cf. A000583, A001511, A035363, A053251, A344619, A357136, A357182, A357627, A357628, A357629, A357633.

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[2n],EvenQ[Length[#]]&&halfats[#]==0&]],{n,0,9}]

a(n) = {my(v, res); if(n < 3, return(1 - bitand(n,1))); res = 0; v = vector(2*n, i, binomial(n-1,i-1)); forstep(i = 4, 2*n, 2, lp = i\4 * 2; rp = i - lp; res += v[lp] * v[rp]; ); res } \\ David A. Corneth, Oct 13 2022

More terms from Alois P. Heinz, Oct 12 2022

cp's OEIS Frontend

A357641 Number of integer compositions of 2n whose half-alternating sum is 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A357621 Half-alternating sum of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A357637 Triangle read by rows where T(n,k) is the number of integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A357182 Number of integer compositions of n with the same length as their alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A357189 Number of integer partitions of n with the same length as alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A357623 Skew-alternating sum of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A357625 Numbers k such that the k-th composition in standard order has half-alternating sum 0.

Views

Author

Keywords

Comments