A350947 -id:A350947 - cp's OEIS frontend

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.

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}]

A351980 Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.

Original entry on oeis.org

1, 6, 84, 126, 140, 210, 490, 525, 686, 875, 1404, 1456, 2106, 2184, 2288, 2340, 3432, 3510, 5460, 6760, 7644, 8190, 8580, 8775, 9100, 9464, 11466, 12012, 12740, 12870, 13650, 14300, 14625, 15808, 18018, 18468, 19110, 19152, 20020, 20672, 21450, 22308, 23712

Offset: 1

Gus Wiseman, Mar 14 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: ()
     6: (2,1)
    84: (4,2,1,1)
   126: (4,2,2,1)
   140: (4,3,1,1)
   210: (4,3,2,1)
   490: (4,4,3,1)
   525: (4,3,3,2)
   686: (4,4,4,1)
   875: (4,3,3,3)
  1404: (6,2,2,2,1,1)
  1456: (6,4,1,1,1,1)
  2106: (6,2,2,2,2,1)
  2184: (6,4,2,1,1,1)
  2288: (6,5,1,1,1,1)
  2340: (6,3,2,2,1,1)

The first condition alone is A349157, counted by A277579.
The second condition alone is A350943, counted by A277579.
There are two other possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
The case of all four statistics equal is A350947, counted by A351978.
These partitions are counted by A351981.
Partitions with as many even as odd parts:
- counted by A045931
- strict case counted by A239241
- ranked by A325698
- conjugate ranked by A350848
- strict conjugate case counted by A352129
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350841, A350849, A350941, 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[1000],Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&&Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]&]

Closed under A122111 (conjugation).
Intersection of A349157 and A350943.
A257992(a(n)) = A344616(a(n)).
A257991(a(n)) = A350847(a(n)).

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}]

A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      13         15         18         20           22
   ------------------------------------------------------------------
    (2,1)  (6,5,2)    (10,5)     (12,6)     (12,7,1)     (12,8,2)
           (6,4,2,1)  (6,4,3,2)  (8,7,3)    (8,5,4,3)    (8,6,5,3)
                      (6,5,3,1)  (8,5,3,2)  (8,6,4,2)    (8,7,5,2)
                                 (8,6,3,1)  (8,7,4,1)    (12,7,2,1)
                                            (8,6,3,2,1)  (8,6,4,3,1)
                                                         (8,7,4,2,1)

This is the strict case of A045931, ranked by A350848 (zeros of A350941).
The conjugate version is A239241, non-strict A045931 (ranked by A325698).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, ranked by A349157, strict A352131.
- A277103, ranked by A350944.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.

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

A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 2    7        9        13        14         15         16
   --------------------------------------------------------------------
    (2)  (6,1)    (8,1)    (12,1)    (14)       (14,1)     (16)
         (4,2,1)  (4,3,2)  (6,4,3)   (6,5,3)    (6,5,4)    (8,5,3)
                  (6,2,1)  (8,3,2)   (10,3,1)   (8,4,3)    (12,3,1)
                           (10,2,1)  (6,4,3,1)  (10,3,2)   (6,5,4,1)
                                     (8,3,2,1)  (12,2,1)   (8,4,3,1)
                                                (6,5,3,1)  (10,3,2,1)
                                                           (6,4,3,2,1)

This is the strict case of A277579, ranked by A350943 (zeros of A350942).
The conjugate version is A352131, non-strict A277579 (ranked by A349157).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944, strict new.
- A350948, ranked by A350945, strict new.
There are three double-pairings of statistics:
- A351976, ranked by A350949, strict A010054?
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980. strict A014105?
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.

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

A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      10         14         18         21             24
   ----------------------------------------------------------------------
    (2,1)  (6,4)      (8,6)      (10,8)     (11,10)        (8,7,5,4)
           (4,3,2,1)  (5,4,3,2)  (6,5,4,3)  (8,6,4,3)      (9,8,4,3)
                      (6,5,2,1)  (7,6,3,2)  (8,7,4,2)      (10,8,4,2)
                                 (8,7,2,1)  (10,8,2,1)     (10,9,3,2)
                                            (6,5,4,3,2,1)  (11,10,2,1)
                                                           (8,6,4,3,2,1)

This is the strict case of A277579, ranked by A349157 (zeros of A350849).
The conjugate version is A352130, non-strict A277579 (ranked by A350943).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350942, A350950, A350951.

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

A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 2, 2, 5, 2, 5, 4, 6, 7, 7, 8, 8, 9, 9, 13, 9, 14, 12, 20, 13, 25, 17, 33, 23, 40, 26, 50, 33, 59, 39, 68, 45, 84, 58, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      18         22          28           31              32
   -----------------------------------------------------------------------
    (2,1)  (8,5,3,2)  (8,6,5,3)   (12,7,5,4)   (10,7,5,4,3,2)  (12,8,7,5)
           (8,6,3,1)  (8,7,5,2)   (12,8,5,3)   (10,7,6,5,2,1)  (12,9,7,4)
                      (12,7,2,1)  (12,9,5,2)   (10,8,5,4,3,1)  (16,9,4,3)
                                  (16,9,2,1)   (10,9,6,3,2,1)  (12,10,7,3)
                                  (12,10,5,1)                  (12,11,7,2)
                                                               (16,11,4,1)

The first condition is A239241, non-strict A045931 (ranked by A325698).
This is the strict version of A351977, ranked by A350946.
The second condition is A352129, non-strict A045931 (ranked by A350848).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, strict A352131.
- A277103, ranked by A350944, strict A000700.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are two other double-pairings of statistics:
- A351976, ranked by A350949.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A014105, A088218, A098123, A195017, A236559, A236914, A241638, A325700, A350839, A350941.

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

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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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A351980 Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.

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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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A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

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A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

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A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

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A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

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