A120452 -id:A120452 - cp's OEIS frontend

A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88

Offset: 1

Gus Wiseman, Jul 04 2021

nonn

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

			The sequence of terms together with the corresponding compositions begins:
     0: ()           19: (3,1,1)        40: (2,4)
     1: (1)          20: (2,3)          41: (2,3,1)
     2: (2)          21: (2,2,1)        42: (2,2,2)
     3: (1,1)        22: (2,1,2)        43: (2,2,1,1)
     4: (3)          24: (1,4)          44: (2,1,3)
     6: (1,2)        26: (1,2,2)        46: (2,1,1,2)
     7: (1,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
     8: (4)          28: (1,1,3)        48: (1,5)
    10: (2,2)        30: (1,1,1,2)      50: (1,3,2)
    11: (2,1,1)      31: (1,1,1,1,1)    51: (1,3,1,1)
    12: (1,3)        32: (6)            52: (1,2,3)
    13: (1,2,1)      35: (4,1,1)        53: (1,2,2,1)
    14: (1,1,2)      36: (3,3)          54: (1,2,1,2)
    15: (1,1,1,1)    37: (3,2,1)        55: (1,2,1,1,1)
    16: (5)          38: (3,1,2)        56: (1,1,4)

The version for prime indices is A000027, counted by A000041.
These compositions are counted by A116406.
The case of non-Heinz numbers of partitions is A119899, counted by A344608.
The version for Heinz numbers of partitions is A344609, counted by A344607.
These are the positions of terms >= 0 in A344618.
The version for unreversed alternating sum is A345913.
The opposite (k <= 0) version is A345916.
The strict (k > 0) case is A345918.
The complement is A345920, counted by A294175.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
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, A027187, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.

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

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).

A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 01 2021

nonn

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 prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.

The version for standard compositions is A209281(n+1) (even: A346633).
Subtracting the even version gives A316524 (reverse: A344616).
The even version is A346698.
The reverse version is A346699.
The even reverse version is A346700.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A000302 counts compositions with odd alternating sum, ranked by A053738.
A001414 adds up prime factors, row sums of A027746.
A029837 adds up parts of standard compositions (alternating: A124754).
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Partition[Append[primeMS[n],0],2]],{n,100}]

a(n) = A056239(n) - A346698(n).
a(n) = A316524(n) + A346698(n).
a(n odd omega) = A346699(n).
a(n even omega) = A346700(n).
A344616(n) = A346699(n) - A346700(n).

A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 4, 3, 2, 0, 2, 0, 1, 4, 5, 0, 3, 3, 6, 2, 1, 0, 2, 0, 2, 5, 7, 4, 3, 0, 8, 6, 4, 0, 2, 0, 1, 2, 9, 0, 2, 4, 3, 7, 1, 0, 4, 5, 5, 8, 10, 0, 4, 0, 11, 2, 3, 6, 2, 0, 1, 9, 3, 0, 3, 0, 12, 3, 1, 5, 2, 0, 2, 4, 13, 0, 5, 7, 14, 10, 6, 0, 5, 6, 1, 11, 15, 8, 4, 0, 4, 2, 4, 0, 2, 0, 7, 3

Offset: 1

Gus Wiseman, Aug 01 2021

nonn

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 prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 = 4.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 3 + 4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Subtracting from the odd version gives A316524 (reverse: A344616).
The version for standard compositions is A346633 (odd: A209281(n+1)).
The odd version is A346697.
The even reverse version is A346699.
The reverse version is A346700.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A001414 adds up prime factors, row-sums of A027746.
A029837 adds up parts of standard compositions (alternating: A124754).
A056239 adds up prime indices, row-sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A000097, A025047, A035363, A120452, A344617, A344653, A344654, A345957, A345958.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Last/@Partition[Append[primeMS[n],0],2]],{n,100}]

A346698(n) = if(1==n,0,my(f=factor(n),s=0,p=0); for(k=1,#f~,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ Antti Karttunen, Nov 30 2021

a(n) = A056239(n) - A346697(n).
a(n) = A346697(n) - A316524(n).
a(n even omega) = A346699(n).
a(n odd omega) = A346700(n).
A344616(n) = A346699(n) - A346700(n).

Data section extended up to 105 terms by Antti Karttunen, Nov 30 2021

A152146 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 1, 0, 3, 3, 2, 1, 1, 0, 3, 5, 3, 2, 1, 1, 0, 4, 6, 5, 3, 2, 1, 1, 0, 4, 9, 7, 5, 3, 2, 1, 1, 0, 5, 11, 11, 7, 5, 3, 2, 1, 1, 0, 5, 15, 14, 11, 7, 5, 3, 2, 1, 1, 0, 6, 18, 20, 15, 11, 7, 5, 3, 2, 1, 1, 0, 6, 23, 26, 22, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 0

R. J. Mathar, Sep 25 2009, indices corrected Jul 09 2012

In both this and A152157, reading columns downwards "converges" to A000041.

Also the number of strict integer partitions of 2n with alternating sum 2k. Also the number of normal integer partitions of 2n of which 2k parts are odd, where a partition is normal if it covers an initial interval of positive integers. - Gus Wiseman, Jun 20 2021

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  1   1
  0  2  2   1   1
  0  3  3   2   1   1
  0  3  5   3   2   1   1
  0  4  6   5   3   2   1  1
  0  4  9   7   5   3   2  1  1
  0  5 11  11   7   5   3  2  1  1
  0  5 15  14  11   7   5  3  2  1  1
  0  6 18  20  15  11   7  5  3  2  1  1
  0  6 23  26  22  15  11  7  5  3  2  1  1
  0  7 27  35  29  22  15 11  7  5  3  2  1  1
  0  7 34  44  40  30  22 15 11  7  5  3  2  1 1
  0  8 39  58  52  42  30 22 15 11  7  5  3  2 1 1
  0  8 47  71  70  55  42 30 22 15 11  7  5  3 2 1 1
  0  9 54  90  89  75  56 42 30 22 15 11  7  5 3 2 1 1
  0  9 64 110 116  97  77 56 42 30 22 15 11  7 5 3 2 1 1
  0 10 72 136 146 128 100 77 56 42 30 22 15 11 7 5 3 2 1 1
From _Gus Wiseman_, Jun 20 2021: (Start)
For example, row n = 6 counts the following partitions (B = 11):
  (75)  (3333)  (333111)  (33111111)  (3111111111)  (111111111111)
  (93)  (5331)  (531111)  (51111111)
  (B1)  (5511)  (711111)
        (7311)
        (9111)
The corresponding strict partitions are:
  (7,5)      (8,4)      (9,3)    (10,2)   (11,1)  (12)
  (6,5,1)    (5,4,3)    (7,3,2)  (9,2,1)
  (5,4,2,1)  (6,4,2)    (8,3,1)
             (7,4,1)
             (6,3,2,1)
The corresponding normal partitions are:
  43221    33321     3321111    321111111   21111111111  111111111111
  322221   332211    32211111   2211111111
  2222211  432111    222111111
           3222111
           22221111
(End)

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

Cf. A035294 (row sums), A107379, A152140, A152157.
Column k = 1 is A004526.
Column k = 2-8 is A026810 - A026816.
The non-strict version is A239830.
The reverse non-strict version is A344610.
The reverse version is A344649
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A006330, A027187, A116406, A120452, A239829, A343941, A344607, A344608, A344609, A344650, A344651, A344739, A344741.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(2*n, 2*n-1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 21 2021

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&ats[#]==k&]],{n,0,30,2},{k,0,n,2}] (* Gus Wiseman, Jun 20 2021 *)

T(n,k) = A152140(2n,2k).

A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 03 2021

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 partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 5 + 3 + 1 = 9.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 4 + 3 + 1 = 8.

The version for standard compositions is A209281(n+1) (even: A346633).
Subtracting the even version gives A344616 (non-reverse: A316524).
The even version is A346700.
The non-reverse version (multisets instead of partitions) is A346697.
The even non-reverse version is A346698.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A124754, A329888, A341446, A344617, A345957, A345958, A346701, A346703.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Partition[Append[Reverse[primeMS[n]],0],2]],{n,100}]

a(n) = A056239(A346701(n)).
a(n) = A056239(n) - A346700(n).
a(n) = A344616(n) + A346700(n).
a(n odd omega) = A346697(n).
a(n even omega) = A346698(n).
A316524(n) = A346697(n) - A346698(n).

A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 3, 3, 2, 1, 0, 1, 0, 1, 4, 4, 3, 2, 1, 0, 1, 0, 1, 5, 6, 4, 3, 2, 1, 0, 1, 0, 1, 7, 7, 6, 4, 3, 2, 1, 0, 1, 0, 1, 8, 10, 8, 6, 4, 3, 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 2n into an odd number of parts whose conjugate has exactly 2k odd parts.

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

			Triangle begins:
   1
   0   1
   0   0   1
   0   1   0   1
   0   1   1   0   1
   0   1   2   1   0   1
   0   1   3   2   1   0   1
   0   1   3   3   2   1   0   1
   0   1   4   4   3   2   1   0   1
   0   1   5   6   4   3   2   1   0   1
   0   1   7   7   6   4   3   2   1   0   1
   0   1   8  10   8   6   4   3   2   1   0   1
   0   1  10  13  12   8   6   4   3   2   1   0   1
   0   1  11  18  15  12   8   6   4   3   2   1   0   1
   0   1  14  22  21  16  12   8   6   4   3   2   1   0   1
   0   1  15  29  27  23  16  12   8   6   4   3   2   1   0   1
Row n = 8 counts the following partitions (empty columns indicated by dots):
  .  (8,7,1)  (7,6,3)      (7,5,4)   (9,4,3)   (11,3,2)  (13,2,1)  .  (16)
              (8,6,2)      (8,5,3)   (10,4,2)  (12,3,1)
              (9,6,1)      (9,5,2)   (11,4,1)
              (6,4,3,2,1)  (10,5,1)
Row n = 9 counts the following partitions (empty columns indicated by dots, A..I = 10..18):
  .  981   873     765     954   B43   D32   F21   .  I
           972     864     A53   C42   E31
           A71     963     B52   D41
           65421   A62     C51
           75321   B61
                   84321

The non-reversed version is A152146.
The non-reversed non-strict version is A239830.
Column k = 2 is A343941.
The non-strict version is A344610.
Row sums are A344650.
Right half of even-indexed rows of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A003242, A006330, A027187, A114121, A116406, A239829, A344607, A344608, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,30,2},{k,0,n,2}]

A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 3, 4, 4, 4, 8, 11, 11, 11, 20, 27, 29, 31, 48, 65, 70, 74, 109, 145, 160, 172, 238, 314, 345, 372, 500, 649, 721, 782, 1019, 1307, 1451, 1577, 2015, 2552, 2841, 3098, 3885, 4867, 5418, 5914, 7318, 9071, 10109, 11050

Offset: 0

Gus Wiseman, Jun 26 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)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. By conjugation, this is also (-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 n of even rank with the same number of odd parts as their conjugate.

			The a(5) = 1 through a(12) = 11 partitions:
  (311)  (321)  (43)    (44)    (333)    (541)    (65)      (66)
                (2221)  (332)   (531)    (4321)   (4322)    (552)
                (4111)  (2222)  (32211)  (32221)  (4331)    (4332)
                        (4211)  (51111)  (52111)  (4421)    (4422)
                                                  (6311)    (4431)
                                                  (222221)  (6411)
                                                  (422111)  (33222)
                                                  (611111)  (53211)
                                                            (222222)
                                                            (422211)
                                                            (621111)

The non-reverse version is A277103.
Comparing even parts to odd conjugate parts gives A277579.
Comparing signs only gives A340601.
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 rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A006330, A027187, A027193, A236559, A239829, A257991, A344607, A344608, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],sats[#]==sats[conj[#]]&]],{n,0,15}]

A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.

Original entry on oeis.org

0, 9, 11, 14, 130, 133, 135, 138, 141, 143, 148, 153, 155, 158, 168, 177, 179, 182, 188, 208, 225, 227, 230, 236, 248, 2052, 2057, 2059, 2062, 2066, 2069, 2071, 2074, 2077, 2079, 2084, 2089, 2091, 2094, 2098, 2101, 2103, 2106, 2109, 2111, 2120, 2129, 2131

Offset: 1

Gus Wiseman, Oct 29 2021

nonn

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.

			The terms together with their binary indices begin:
    0: ()
    9: (3,1)
   11: (2,1,1)
   14: (1,1,2)
  130: (6,2)
  133: (5,2,1)
  135: (5,1,1,1)
  138: (4,2,2)
  141: (4,1,2,1)
  143: (4,1,1,1,1)
  148: (3,2,3)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)

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

The unordered case (partitions) is counted by A000712, reverse A006330.
These compositions are counted by A262977.
Except for 0, a subset of A345917 (which is itself a subset of A345913).
A000346 = even-length compositions with alt sum != 0, complement A001700.
A011782 counts compositions.
A025047 counts wiggly compositions, ranked by A345167.
A034871 counts compositions of 2n with alternating sum 2k.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
A345197 counts compositions by length and alternating sum.
Cf. A000984, A002458, A004331, A005810, A224274.
Cf. A008549, A013777, A027306, A058622, A088218, A114121, A120452, A126869, A163493, A294175, A344604.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==2*ats[stc[#]]&]

A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 29, 54, 96, 165, 275, 449, 716, 1123, 1732, 2635, 3955, 5871, 8620, 12536, 18065, 25821, 36617, 51560, 72105, 100204, 138417, 190134, 259772, 353134, 477734, 643354, 862604, 1151773, 1531738, 2029305, 2678650, 3523378, 4618835, 6035240, 7861292

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

Conjecture: a(n) >= A236914.
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 a(n) is the number of even-length partitions of 2n with at least one odd conjugate part. By conjugation, this is also the number of partitions of 2n with greatest part even and at least one odd part.
The alternating sum of a partition is never < 0, so the non-reverse version is A000004.

			The a(2) = 1 through a(5) = 15 partitions:
  (31)  (42)    (53)      (64)
        (51)    (62)      (73)
        (3111)  (71)      (82)
                (3221)    (91)
                (4211)    (3331)
                (5111)    (4222)
                (311111)  (4321)
                          (5221)
                          (5311)
                          (6211)
                          (7111)
                          (322111)
                          (421111)
                          (511111)
                          (31111111)

The ordered version (compositions not partitions) appears to be A008549.
The Heinz numbers are A119899 /\ A300061.
Even bisection of A344608.
The complementary partitions of 2n are counted by A344611.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001523 counts unimodal compositions (partial sums: A174439).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A006330, A027187, A239830, A343941, A344604, A344607, A344650, A344651, A344654.

sats[y_] := Sum[(-1)^(i - Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30,2}]

a(n) = A058696(n) - A344611(n).

a(n) = sum of left half of even-indexed rows of A344612.

More terms from Bert Dobbelaere, Jun 12 2021

cp's OEIS Frontend

A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

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

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A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

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A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.

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A152146 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of 2n into 2k odd parts.

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A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

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A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

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A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

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