A236559 -id:A236559 - cp's OEIS frontend

A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

1, 2, 6, 9, 10, 12, 15, 18, 20, 30, 35, 49, 54, 55, 56, 70, 75, 77, 81, 84, 88, 90, 98, 108, 110, 112, 125, 132, 135, 143, 154, 162, 168, 169, 176, 180, 187, 210, 221, 260, 264, 270, 286, 294, 315, 323, 330, 338, 340, 350, 361, 363, 364, 374, 391, 416, 420

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)
   6: (2,1)
   9: (2,2)
  10: (3,1)
  12: (2,1,1)
  15: (3,2)
  18: (2,2,1)
  20: (3,1,1)
  30: (3,2,1)
  35: (4,3)
  49: (4,4)
  54: (2,2,2,1)

These partitions are counted by A277103.
The even rank case is A345196.
The conjugate version is A350943, counted by A277579.
These are the positions of 0's in A350951, even 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: # even conj parts = # odd conj parts, counted by A045931.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000700, A027187, A027193, A103919, A236559, A350942.

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[#]],?OddQ]==Count[primeMS[#],?OddQ]&]

A257991(a(n)) = A344616(a(n)).

A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

Original entry on oeis.org

1, 3, 6, 7, 13, 14, 18, 19, 26, 27, 29, 36, 37, 38, 42, 43, 53, 54, 58, 61, 63, 70, 71, 74, 78, 79, 84, 86, 89, 101, 105, 106, 107, 113, 114, 117, 122, 126, 130, 131, 139, 140, 142, 151, 156, 158, 162, 163, 171, 173, 174, 178, 181, 190, 193, 195, 199, 202, 210

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: ()
   3: (2)
   6: (2,1)
   7: (4)
  13: (6)
  14: (4,1)
  18: (2,2,1)
  19: (8)
  26: (6,1)
  27: (2,2,2)
  29: (10)
  36: (2,2,1,1)
  37: (12)
  38: (8,1)
  42: (4,2,1)
For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 is in the sequence.

These partitions are counted by A277579.
The conjugate version is A349157, also counted by A277579.
These are the positions of 0's in A350942.
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.
  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, 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[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]&]

A350847(a(n)) = A257991(a(n)).

A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, 0, -3, 0, 3, 1, 1, 0, 0, 0, -1, -1, -2, 0, 1, 0, 0, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, -2, -1, 1, 1, 2, -3, 0, 0, 0, 0, -1, 1, -1, 3, -1, -2, 0, 0, 0, -1, -1, 1, 1, 0, 0, 0, 1, -3, 1, 1, 0

Offset: 1

Gus Wiseman, Mar 14 2022

sign

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 prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 2 - 1 = 1.

The version comparing even with odd parts is A195017.
The version comparing even with odd conjugate parts is A350849.
The version comparing even conjugate with odd conjugate parts is A350941.
The version comparing odd with even conjugate parts is A350942.
Positions of 0's are A350945, counted by A350948.
The version comparing odd with odd conjugate parts is A350951.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are five other possible pairings of statistics:
- 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.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Cf. A028260, A088218, A098123, A236559, A239241, A241638, A325700, A347450.

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]}]];
Table[Count[primeMS[n],?EvenQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

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

a(A122111(n)) = -a(n), where A122111 represents partition conjugation.

A350951 Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, -2, 2, -2, 0, -4, 2, 0, 0, -4, 0, -6, -2, 0, 4, -6, 0, -8, 0, -2, -2, -8, 2, 2, -4, -2, -2, -10, 0, -10, 4, -2, -4, 0, 2, -12, -6, -4, 2, -12, -2, -14, -2, -2, -6, -14, 2, 0, 2, -4, -4, -16, 0, 0, 0, -6, -8, -16, 2, -18, -8, -4, 6, -2, -2, -18, -4, -6, 0

Offset: 1

Gus Wiseman, Mar 14 2022

sign

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.

All terms are even.

			The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 1 - 5 = -4.

The version comparing even with odd parts is A195017.
The version comparing even with odd conjugate parts is A350849.
The version comparing even conjugate with odd conjugate parts is A350941.
The version comparing odd with even conjugate parts is A350942.
Positions of 0's are A350944, even rank case A345196, counted by A277103.
The version comparing even with even conjugate parts is A350950.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are five other possible pairings of statistics:
- 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.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A103919 counts partitions by number of odd parts.
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Cf. A026424, A028260, A053738, A098123, A130780, A171966, A236559, A236914, A239241, A241638, A325700.

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]}]];
Table[Count[primeMS[n],?OddQ]-Count[conj[primeMS[n]],?OddQ],{n,100}]

a(n) = A257991 - A344616(n).

a(A122111(n)) = -a(n), where A122111 represents partition conjugation.

A351976 Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 6, 9, 11, 11, 16, 21, 22, 24, 31, 41, 46, 48, 64, 82, 91, 98, 120, 155, 175, 188, 237, 297, 329, 357, 437, 544, 607, 658, 803, 987, 1098, 1196, 1432, 1749, 1955, 2126, 2541, 3071, 3417, 3729, 4406, 5291, 5890, 6426

Offset: 0

Gus Wiseman, Mar 14 2022

nonn

			The a(n) partitions for selected n:
n = 3     8       11        12        15          16
   ----------------------------------------------------------
    (21)  (332)   (4322)    (4332)    (4443)      (4444)
          (4211)  (4331)    (4422)    (54321)     (53332)
                  (4421)    (4431)    (632211)    (55222)
                  (611111)  (53211)   (633111)    (55411)
                            (621111)  (642111)    (633211)
                                      (81111111)  (642211)
                                                  (643111)
                                                  (7321111)
                                                  (82111111)

The first condition alone is A277103, ranked by A350944, strict A000700.
The second condition alone is A350948, ranked by A350945.
These partitions are ranked by A350949.
A000041 counts integer partitions.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other possible pairings of statistics:
- A045931: # even = # odd, ranked by A325698, strict A239241.
- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
There are two other possible double-pairings of statistics:
- A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.
- A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A088218, A098123, A130780, A171966, A236559, A236914, A241638, A350849, A350941, A350942, A350950, A350951.

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

A351977 Number of integer partitions of n with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 2, 4, 2, 1, 6, 6, 7, 9, 11, 10, 13, 17, 17, 21, 28, 36, 35, 41, 58, 71, 72, 90, 106, 121, 142, 178, 191, 216, 269, 320, 344, 400, 486, 564, 633, 734, 867, 991, 1130, 1312, 1509, 1702, 1978, 2288, 2582, 2917, 3404

Offset: 0

Gus Wiseman, Mar 14 2022

nonn

			The a(n) partitions for selected n (A..C = 10..12):
n = 3     9         15            18          20
   ----------------------------------------------------------
    (21)  (63)      (A5)          (8433)      (8543)
          (222111)  (632211)      (8532)      (8741)
                    (642111)      (8631)      (C611)
                    (2222211111)  (43322211)  (43332221)
                                  (44322111)  (44432111)
                                  (44421111)  (84221111)
                                              (422222111111)

The first condition alone is A045931, ranked by A325698, strict A239241.
The second condition alone is A045931, ranked by A350848, strict A352129.
These partitions are ranked by A350946.
The strict case is A352128.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four additional pairings of statistics:
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
- A277103: # odd = # odd conj, ranked by A350944, strict A000700.
- A350948: # even = # even conj, ranked by A350945.
There are two additional double-pairings of statistics:
- A351981, ranked by A351980.
- A351976, ranked by A350949.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000041, A000070, A088218, A098123, A130780, A171966, A195017, A236559, A236914, A241638, A350849.

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

A351978 Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 6, 1, 3, 1, 8, 5, 3, 5, 7, 14, 2, 13, 9, 28, 5, 22, 26, 44, 17, 30, 60, 59, 42, 41, 120, 84, 84, 66, 204, 143, 144, 131, 325, 268, 226, 261, 486, 498, 344, 488, 739, 874

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) partitions for selected n (A = 10):
n = 3    12     19       21       23       24         27
   --------------------------------------------------------------
    21   4332   633322   643332   644333   84332211   655443
         4431   643321   654321   654332   84441111   655542
                644311   665211   654431   85322211   665541
                653221            655322   86322111   666333
                654211            655421   86421111   666531
                664111            664331              A522221111
                                  665321              A622211111
                                  666311

The strict case appears to be the indicator function for A014105.
These partitions are ranked by A350947.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are six pairings of statistics:
- A045931: # of even parts = # of odd parts:
  - ordered A098123
  - strict A239241
  - ranked by A325698
- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277103: # odd = # odd conj, ranked by A350944, strict A000700.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
- A350948: # even = # even conj, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946.
- A351981, ranked by A351980.
A000041 counts integer partitions, strict A000009.
A103919 and A116482 count partitions by sum and number of odd/even parts.
A195017 = # of even parts - # of odd parts.
Cf. A000070, A122111, A130780, A171966, A236559, A236914, A350849, A350941, A350942, A350950, A350951.

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

A351981 Number of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 2, 2, 4, 2, 1, 6, 8, 7, 9, 13, 14, 15, 19, 21, 23, 32, 40, 41, 45, 66, 81, 80, 96, 124, 139, 160, 194, 221, 246, 303, 360, 390, 446, 546, 634, 703, 810, 971, 1115, 1250, 1448, 1685, 1910

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) partitions for selected n:
n = 3    9      15       18       19       20         21
   -----------------------------------------------------------
    21   4221   622221   633222   633322   644321     643332
         4311   632211   643221   643321   653321     654321
                642111   643311   644221   654221     665211
                651111   644211   644311   654311     82222221
                         653211   653221   82222211   83222211
                         663111   653311   84221111   84222111
                                  654211   86111111   85221111
                                  664111              86211111
                                                      87111111
For example, the partition (6,6,3,1,1,1) has conjugate (6,3,3,2,2,2), and has 2 even, 4 odd, 4 even conjugate, and 2 odd conjugate parts, so is counted under a(18).

The first condition alone is A277579, ranked by A349157.
The second condition alone is A277579, ranked by A350943.
These partitions are ranked by A351980.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931: # of even parts = # of odd parts:
  - conjugate also A045931
  - ordered A098123
  - strict A239241
  - ranked by A325698
  - conjugate ranked by A350848
- A277103: # of odd parts = # of odd conjugate parts, ranked by A350944.
- A350948: # of even parts = # of even conjugate parts, ranked by A350945.
There are two other double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A088218, A122111, A130780, A171966, A195017, A236559, A236914, A350849, A350942.

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[#],?OddQ]&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A347444 Number of odd-length integer partitions of n with integer alternating product.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398

Offset: 0

Gus Wiseman, Sep 14 2021

nonn

We define the alternating product of a sequence (y_1, ... ,y_k) to be the Product_i y_i^((-1)^(i-1)).

The reverse version (integer reverse-alternating product) is the same.

			The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)        (9)
            (111)  (211)  (221)    (222)    (322)      (332)      (333)
                          (311)    (411)    (331)      (422)      (441)
                          (11111)  (21111)  (421)      (611)      (522)
                                            (511)      (22211)    (621)
                                            (22111)    (41111)    (711)
                                            (31111)    (2111111)  (22221)
                                            (1111111)             (32211)
                                                                  (33111)
                                                                  (42111)
                                                                  (51111)
                                                                  (2211111)
                                                                  (3111111)
                                                                  (111111111)

The reciprocal version is A035363.
Allowing any alternating product gives A027193.
The multiplicative version (factorizations) is A347441.
Allowing any length gives A347446, reverse A347445.
Allowing any length and alternating product > 1 gives A347448.
Allowing any reverse-alternating product > 1 gives A347449.
Ranked by A347453.
The even-length instead of odd-length version is A347704.
A000041 counts partitions.
A000302 counts odd-length compositions, ranked by A053738.
A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A027187 counts partitions of even length, strict A067661.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

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}]

cp's OEIS Frontend

A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

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A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

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A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.

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A350951 Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.

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A351976 Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.

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A351977 Number of integer partitions of n with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

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A351978 Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.

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A351981 Number of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.

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A347444 Number of odd-length integer partitions of n with integer alternating product.

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