A344651 -id:A344651 - cp's OEIS frontend

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.

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.

A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 44, 45, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jul 10 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
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.
Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.

			The initial terms and the corresponding compositions:
     1: (1)        20: (2,3)          35: (4,1,1)
     2: (2)        21: (2,2,1)        37: (3,2,1)
     4: (3)        22: (2,1,2)        38: (3,1,2)
     5: (2,1)      23: (2,1,1,1)      39: (3,1,1,1)
     6: (1,2)      24: (1,4)          40: (2,4)
     7: (1,1,1)    25: (1,3,1)        42: (2,2,2)
     8: (4)        26: (1,2,2)        44: (2,1,3)
     9: (3,1)      27: (1,2,1,1)      45: (2,1,2,1)
    11: (2,1,1)    28: (1,1,3)        47: (2,1,1,1,1)
    12: (1,3)      29: (1,1,2,1)      48: (1,5)
    14: (1,1,2)    30: (1,1,1,2)      49: (1,4,1)
    16: (5)        31: (1,1,1,1,1)    51: (1,3,1,1)
    17: (4,1)      32: (6)            52: (1,2,3)
    18: (3,2)      33: (5,1)          54: (1,2,1,2)
    19: (3,1,1)    34: (4,2)          56: (1,1,4)

The version for Heinz numbers of partitions is A000037.
These compositions are counted by A058622.
These are the positions of terms != 0 in A124754.
The complement (k = 0) is A344619.
The positive (k > 0) version is A345917 (reverse: A345918).
The negative (k < 0) version is A345919 (reverse: A345920).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A008549, A025047, A032443, A034871, A114121, A163493, A236913, A344609, A344651, A345908.

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],ats[stc[#]]!=0&]

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}]

A344650 Number of strict odd-length integer partitions of 2n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 44, 61, 82, 111, 148, 195, 256, 334, 432, 557, 713, 908, 1152, 1455, 1829, 2291, 2859, 3554, 4404, 5440, 6697, 8222, 10066, 12288, 14964, 18176, 22023, 26625, 32117, 38656, 46432, 55661, 66592, 79523, 94793, 112792, 133984

Offset: 0

Gus Wiseman, Jun 05 2021

nonn

Also the number of strict integer partitions of 2n with reverse-alternating sum >= 0.

Also the number of reversed strict integer partitions of 2n with alternating sum >= 0.

			The a(1) = 1 through a(8) = 16 partitions:
  (2)  (4)  (6)      (8)      (10)     (12)     (14)      (16)
            (3,2,1)  (4,3,1)  (5,3,2)  (5,4,3)  (6,5,3)   (7,5,4)
                     (5,2,1)  (5,4,1)  (6,4,2)  (7,4,3)   (7,6,3)
                              (6,3,1)  (6,5,1)  (7,5,2)   (8,5,3)
                              (7,2,1)  (7,3,2)  (7,6,1)   (8,6,2)
                                       (7,4,1)  (8,4,2)   (8,7,1)
                                       (8,3,1)  (8,5,1)   (9,4,3)
                                       (9,2,1)  (9,3,2)   (9,5,2)
                                                (9,4,1)   (9,6,1)
                                                (10,3,1)  (10,4,2)
                                                (11,2,1)  (10,5,1)
                                                          (11,3,2)
                                                          (11,4,1)
                                                          (12,3,1)
                                                          (13,2,1)
                                                          (6,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

The Heinz numbers are the intersection of A030059 and A300061.
Allowing even length gives A035294 (non-strict: A058696).
Even bisection of A067659.
The opposite type of strict partition (even length and odd sum) is A343942.
The non-strict version is A236559 or A344611.
Row sums of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sums of standard compositions (reverse: A344618).
A152146 interleaved with A152157 counts strict partitions by sum and alternating sum.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A343941 counts strict partitions of 2n with reverse-alternating sum 4.
A344604 counts wiggly compositions with twins.
A344739 counts strict partitions by sum and reverse-alternating sum.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A027187, A114121, A239829, A239830, A344607, A344609, A344610, A344651, A344654.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30,2}]

Sum of odd-indexed terms in row 2n of A008289.

a(n) = A067659(2n).

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.

A350945 Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 36, 38, 39, 41, 44, 47, 56, 57, 58, 59, 66, 67, 68, 73, 74, 75, 80, 83, 84, 86, 87, 92, 96, 97, 102, 103, 104, 106, 109, 111, 120, 122, 124, 125, 127, 128, 129, 137, 138, 142, 144, 149, 152, 156

Offset: 1

Gus Wiseman, Jan 28 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 terms together with their prime indices begin:
   1: ()
   2: (1)
   5: (3)
   6: (2,1)
   8: (1,1,1)
   9: (2,2)
  11: (5)
  14: (4,1)
  17: (7)
  20: (3,1,1)
  21: (4,2)
  23: (9)
  24: (2,1,1,1)

These partitions are counted by A350948.
These are the positions of 0's in A350950.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  A325698: # of even parts = # of odd parts, counted by A045931.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # of even conj parts = # of odd conj parts, counted by A045931.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, `A350942, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[conj[primeMS[#]],?EvenQ]==Count[primeMS[#],?EvenQ]&]

A257992(a(n)) = A350847(a(n)).

A344739 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with reverse-alternating sum k, with k ranging from -n to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 2, 1, 0, 1

Offset: 0

Gus Wiseman, Jun 05 2021

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of n such that k is equal to (-1)^(m-1) times the number of odd conjugate parts.
By conjugation, T(n,k) is also equal to the number of integer partitions of n covering an initial interval of positive integers such that k is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part.
Also the number of reversed strict integer partitions of n with alternating sum k.

			Triangle begins:
                                      1
                                    0   1
                                  0   0   1
                                0   1   0   1
                              0   1   0   0   1
                            0   1   1   0   0   1
                          0   1   1   0   1   0   1
                        0   1   1   1   0   1   0   1
                      0   1   1   1   0   1   1   0   1
                    0   1   1   1   1   0   2   1   0   1
                  0   1   1   1   2   0   1   2   1   0   1
                0   1   1   1   2   1   0   2   2   1   0   1
              0   1   1   1   2   2   0   1   3   2   1   0   1
            0   1   1   1   2   3   1   0   2   3   2   1   0   1
          0   1   1   1   2   3   3   0   1   3   3   2   1   0   1
        0   1   1   1   2   3   4   1   0   3   4   3   2   1   0   1
      0   1   1   1   2   3   5   3   0   1   4   4   3   2   1   0   1
    0   1   1   1   2   3   5   5   1   0   3   5   4   3   2   1   0   1
  0   1   1   1   2   3   5   6   4   0   1   5   6   4   3   2   1   0   1
For example, the partitions counted by row n = 15 are (empty columns shown as dots, A...F = 10..15):
  .  E1  D2  C3  B4    A5    96    87  .  762    654  843  A32  C21  .  F
                 9321  7431  6432         861    753  942  B31
                       8421  6531         54321  852  A41
                             7521                951

Row sums are A000009.
The non-reverse version is A152146 interleaved with A152157.
The non-strict version is A344612.
The right halves of even-indexed rows are A344649.
The non-reverse non-strict version is the right half of A344651, which is A239830 interleaved with A239829.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A124754 lists alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions of n by positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A003242, A006330, A027187, A103919, A114121, A343941, A344607, A344608, A344650, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,12},{k,-n,n,2}]

A344741 Number of integer partitions of 2n with reverse-alternating sum -2.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 14, 24, 39, 62, 95, 144, 212, 309, 442, 626, 873, 1209, 1653, 2245, 3019, 4035, 5348, 7051, 9229, 12022, 15565, 20063, 25722, 32847, 41746, 52862, 66657, 83768, 104873, 130889, 162797, 201902, 249620, 307789, 378428, 464122, 567721, 692828, 843448

Offset: 0

Gus Wiseman, Jun 08 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part, so a(n) is the number of integer partitions of 2n with exactly two odd parts, neither of which is the greatest.

Also the number of reversed integer partitions of 2n with alternating sum -2.

			The a(2) = 1 through a(6) = 14 partitions:
  (31)  (42)    (53)      (64)        (75)
        (3111)  (3221)    (3331)      (4332)
                (4211)    (4222)      (4431)
                (311111)  (4321)      (5322)
                          (5311)      (5421)
                          (322111)    (6411)
                          (421111)    (322221)
                          (31111111)  (333111)
                                      (422211)
                                      (432111)
                                      (531111)
                                      (32211111)
                                      (42111111)
                                      (3111111111)

The version for -1 instead of -2 is A000070.
The non-reversed negative version is A000097.
The ordered version appears to be A001700.
The version for 1 instead of -2 is A035363.
The whole set of partitions of 2n is counted by A058696.
The strict case appears to be A065033.
The version for -1 instead of -2 is A306145.
The version for 2 instead of -2 is A344613.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A001250, A003242, A006330, A027187, A028260, A344604, A344607, A344608, A344650, A344651, A344654, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]==-2&]],{n,0,30,2}]
- or -
Table[Length[Select[IntegerPartitions[n],EvenQ[Max[#]]&&Count[#,_?OddQ]==2&]],{n,0,30,2}]

More terms from Bert Dobbelaere, Jun 12 2021

A350948 Number of integer partitions of n with as many even parts as even conjugate parts.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 7, 6, 10, 10, 18, 19, 27, 31, 40, 47, 65, 75, 98, 115, 142, 170, 217, 257, 316, 376, 458, 544, 671, 792, 952, 1129, 1351, 1598, 1919, 2259, 2681, 3155, 3739, 4384, 5181, 6064, 7129, 8331, 9764, 11380, 13308, 15477, 18047, 20944

Offset: 0

Gus Wiseman, Mar 14 2022

nonn

			The a(0) = 1 through a(8) = 6 partitions (empty column indicated by dot):
  ()  (1)  .  (3)    (22)  (5)      (42)    (7)        (62)
              (21)         (41)     (321)   (61)       (332)
              (111)        (311)    (2211)  (511)      (521)
                           (2111)           (4111)     (4211)
                           (11111)          (31111)    (32111)
                                            (211111)   (221111)
                                            (1111111)
For example, both (3,2,1,1,1) and its conjugate (5,2,1) have exactly 1 even part, so are counted under a(8).

Comparing even to odd parts gives A045931, ranked by A325698.
The odd version is A277103, even rank case A345196, ranked by A350944.
Comparing even to odd conjugate parts gives A277579, ranked by A349157.
Comparing product of parts to product of conjugate parts gives A325039.
These partitions are ranked by A350945, the zeros of A350950.
A000041 counts integer partitions, strict A000009.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A116482 counts partitions by number of even (or even conjugate) parts.
A122111 represents partition conjugation using Heinz numbers.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A351976: # even = # even conj, # odd = # odd conj, ranked by A350949.
A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.
A351978: # even = # odd = # even conj = # odd conj, ranked by A350947.
A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.
Cf. A027187, A130780, A171966, A195017, A239241, A241638, A344607, A344651, A350848, A350941, A350942, A350943.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Count[#,?EvenQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

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

cp's OEIS Frontend

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.

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A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

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

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A344650 Number of strict odd-length integer partitions of 2n.

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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.

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A350945 Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

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A344739 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with reverse-alternating sum k, with k ranging from -n to n in steps of 2.

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A344741 Number of integer partitions of 2n with reverse-alternating sum -2.

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