A325039 -id:A325039 - cp's OEIS frontend

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

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}]

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)).

A329382 Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 4, 1, 6, 1, 3, 4, 2, 1, 4, 8, 2, 9, 3, 1, 6, 1, 5, 4, 2, 8, 8, 1, 2, 4, 4, 1, 6, 1, 3, 9, 2, 1, 5, 16, 12, 4, 3, 1, 12, 8, 4, 4, 2, 1, 8, 1, 2, 9, 6, 8, 6, 1, 3, 4, 12, 1, 10, 1, 2, 18, 3, 16, 6, 1, 5, 16, 2, 1, 8, 8, 2, 4, 4, 1, 12, 16, 3, 4, 2, 8, 6, 1, 24, 9, 16, 1, 6, 1, 4, 18

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Also the product of parts of the conjugate of the integer partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (3,2) with Heinz number 15 has conjugate (2,2,1) with product a(15) = 4. - Gus Wiseman, Mar 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A000005, A005361, A019565, A034386, A108951, A284001, A331188, A329378, A329605, A329617.
This is the conjugate version of A003963 (product of prime indices).
The solutions to a(n) = A003963(n) are A325040, counted by A325039.
The Heinz number of the conjugate partition is given by A122111.
These are the row products of A321649 and of A321650.
A000700 counts self-conj partitions, ranked by A088902, complement A330644.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and of A296150.
A124010 gives prime signature, sorted A118914, sum A001222.
A238744 gives the conjugate of prime signature, rank A238745.
Cf. A000701, A000720, A001221, A046682, A175508, A290822, A303975, A316524, A324850, A352486-A352491, A353570.

Table[Times @@ FactorInteger[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]][[All, -1]], {n, 105}] (* Michael De Vlieger, Jan 21 2020 *)

A005361(n) = factorback(factor(n)[, 2]); \\ from A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329382(n) = A005361(A108951(n));

A329382(n) = if(1==n,1,my(f=factor(n),e=0,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

a(n) = A005361(A108951(n)).
A329605(n) >= a(n) >= A329617(n) >= A329378(n).
a(A019565(n)) = A284001(n).
From Antti Karttunen, Jan 14 2020: (Start)
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > k3 > ... > kx, then a(n) = e(k1)^(k1-k2) * (e(k1)+e(k2))^(k2-k3) * (e(k1)+e(k2)+e(k3))^(k3-k4) * ... * (e(k1)+e(k2)+...+e(kx))^kx.
a(n) = A000005(A331188(n)) = A329605(A052126(n)).
(End)
a(n) = A003963(A122111(n)). - Gus Wiseman, Mar 27 2022

A118199 Number of partitions of n having no parts equal to the size of their Durfee squares.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 40, 53, 68, 89, 113, 146, 184, 234, 293, 369, 458, 572, 706, 874, 1073, 1320, 1611, 1970, 2393, 2909, 3518, 4255, 5122, 6167, 7394, 8862, 10585, 12637, 15038, 17886, 21213, 25141, 29723, 35112, 41383, 48737, 57278

Offset: 0

Emeric Deutsch, Apr 14 2006

nonn

a(n) = A118198(n,0).
From Gus Wiseman, May 21 2022: (Start)
Also the number of integer partitions of n > 0 that have a fixed point but whose conjugate does not, ranked by A353316. For example, the a(5) = 1 through a(10) = 10 partitions are:
  11111   222      322       422        522         622
          111111   2221      2222       3222        4222
                   1111111   3221       4221        5221
                             22211      22221       22222
                             11111111   32211       32221
                                        222111      42211
                                        111111111   222211
                                                    322111
                                                    2221111
                                                    1111111111
Partitions w/ a fixed point: A001522 (unproved), ranked by A352827 (cf. A352874).
Partitions w/o a fixed point: A064428 (unproved), ranked by A352826 (cf. A352873).
Partitions w/ a fixed point and a conjugate fixed point: A188674, reverse A325187, ranked by A353317.
Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).
(End)

			a(7) = 3 because we have [7] with size of Durfee square 1, [4,3] with size of Durfee square 2 and [3,3,1] with size of Durfee square 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Column k=0 of A118198.
A000041 counts partitions, strict A000009.
A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
A002467 counts permutations with a fixed point, complement A000166.
A064410 counts partitions of crank 0, ranked by A342192.
A115720 and A115994 count partitions by Durfee square, rank stat A257990.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A114088, A300788, A325039, A350839, A352828, A352829, A352832.

g:=1+sum(x^(k^2+k)/(1-x^k)/product((1-x^i)^2,i=1..k-1),k=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..54);
# second Maple program::
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(add(b(k, d) *b(n-d*(d+1)-k, d-1),
                k=0..n-d*(d+1)), d=0..floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[Sum[b[k, d]*b[n-d*(d+1)-k, d-1], {k, 0, n-d*(d+1)}], {d, 0, Floor[Sqrt[n]]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],pq[#]>0&&pq[conj[#]]==0&]],{n,0,30}] (* a(0) = 0, Gus Wiseman, May 21 2022 *)

G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jun 12 2025

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}]

A325045 Number of factorizations of n whose conjugate as an integer partition has no ones.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

After a(1) = 1, a(n) is the number of factorizations of n with at least two factors, the largest two of which are equal.

			The initial terms count the following factorizations:
    1: {}
    4: 2*2
    8: 2*2*2
    9: 3*3
   16: 2*2*2*2
   16: 4*4
   18: 2*3*3
   25: 5*5
   27: 3*3*3
   32: 2*2*2*2*2
   32: 2*4*4
   36: 2*2*3*3
   36: 6*6
   48: 3*4*4
   49: 7*7
   50: 2*5*5
   54: 2*3*3*3
   64: 2*2*2*2*2*2
   64: 2*2*4*4
   64: 4*4*4
   64: 8*8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001055, A001222, A002865, A096276, A114324, A122111, A318950, A319005, A319916, A320322, A321648, A325039, A353645 [= a(n^2)].

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[facs[n],FreeQ[conj[#],1]&]],{n,1,100}]

A325045(n, m=n, facs=List([])) = if(1==n, (0==#facs || (#facs>=2 && facs[1]==facs[2])), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A325045(n/d, d, newfacs))); (s)); \\ Antti Karttunen, May 03 2022

More terms from Antti Karttunen, May 03 2022

A349149 Number of even-length integer partitions of n with at most one odd part in the conjugate partition.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 7, 12, 11, 19, 15, 30, 22, 45, 30, 67, 42, 97, 56, 139, 77, 195, 101, 272, 135, 373, 176, 508, 231, 684, 297, 915, 385, 1212, 490, 1597, 627, 2087, 792, 2714, 1002, 3506, 1255, 4508, 1575, 5763, 1958, 7338, 2436, 9296, 3010, 11732

Offset: 0

Gus Wiseman, Nov 09 2021

nonn

The alternating sum of a partition is equal to the number of odd parts in the conjugate partition, so this sequence counts even-length partitions with alternating sum <= 1.

			The a(2) = 1 through a(9) = 7 partitions:
  11   21   22     32     33       43       44         54
            1111   2111   2211     2221     2222       3222
                          111111   3211     3311       3321
                                   211111   221111     4311
                                            11111111   222111
                                                       321111
                                                       21111111

The case of 0 odd conjugate parts is A000041 up to 0's, ranked by A000290.
The case of 1 odd conjugate part is A000070 up to 0's.
Even bisection of A100824, ranked by A349150.
Ranked by A349151 /\ A028260.
A045931 counts partitions with as many even as odd parts, ranked by A325698.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A122111 is a representation of partition conjugation.
A277103 counts partitions with the same alternating sum as their conjugate.
A277579 counts partitions with as many even parts as odd conjugate parts.
A325039 counts partitions with the same product as their conjugate.
A344610 counts partitions by sum and positive reverse-alternating sum.
A345196 counts partitions with the same rev-alt sum as their conjugate.
Cf. A000097, A000700, A001700, A027187, A027193, A108711, A236559, A236913, A325534, A344607, A344651.

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

a(2n) = A000041(n).

a(2n+1) = A000070(n-1).

A353317 Heinz numbers of integer partitions that have a fixed point and a conjugate fixed point (counted by A188674).

Original entry on oeis.org

2, 9, 15, 18, 21, 30, 33, 36, 39, 42, 51, 57, 60, 66, 69, 72, 78, 84, 87, 93, 102, 111, 114, 120, 123, 125, 129, 132, 138, 141, 144, 156, 159, 168, 174, 175, 177, 183, 186, 201, 204, 213, 219, 222, 228, 237, 240, 245, 246, 249, 250, 258, 264, 267, 275, 276

Offset: 1

Gus Wiseman, May 15 2022

nonn

A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists.

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 and their prime indices begin:
    2: (1)
    9: (2,2)
   15: (3,2)
   18: (2,2,1)
   21: (4,2)
   30: (3,2,1)
   33: (5,2)
   36: (2,2,1,1)
   39: (6,2)
   42: (4,2,1)
   51: (7,2)
   57: (8,2)
   60: (3,2,1,1)
   66: (5,2,1)
   69: (9,2)
   72: (2,2,1,1,1)
   78: (6,2,1)
   84: (4,2,1,1)
For example, the partition (2,2,1,1) with Heinz number 36 has a fixed point at the second position, as does its conjugate (4,2), so 36 is in the sequence.

These partitions are counted by A188674.
Crank: A342192, A352873, A352874; counted by A064410, A064428, A001522.
The strict case is A352829.
Fixed point but no conjugate fixed point: A353316, counted by A118199.
A000700 counts self-conjugate partitions, ranked by A088902.
A002467 counts permutations with a fixed point, complement A000166.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720/A115994 count partitions by their Durfee square, rank stat A257990.
A122111 represents partition conjugation using Heinz numbers.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352826 ranks partitions w/o a fixed point, counted by A064428 (unproved).
A352827 ranks partitions with a fixed point, counted by A001522 (unproved).
Cf. A001222, A065770, A093641, A252464, A325039, A325163, A325169, A352828, A352831, A352832, A352833.

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

A363220 Number of integer partitions of n whose conjugate has the same median.

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 8, 8, 12, 12, 15, 21, 27, 36, 49, 65, 85, 112, 149, 176, 214, 257, 311, 378, 470, 572, 710, 877, 1080, 1322, 1637, 1983, 2416, 2899, 3465, 4107, 4891, 5763, 6820, 8071, 9542, 11289, 13381, 15808, 18710, 22122, 26105, 30737, 36156, 42377

Offset: 1

Gus Wiseman, May 29 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The partition y = (4,3,1,1) has median 2, and its conjugate (4,2,2,1) also has median 2, so y is counted under a(9).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  .  (21)  (22)  (311)  (321)   (511)    (332)     (333)
                             (411)   (4111)   (422)     (711)
                             (3111)  (31111)  (611)     (4221)
                                              (3311)    (4311)
                                              (4211)    (6111)
                                              (5111)    (51111)
                                              (41111)   (411111)
                                              (311111)  (3111111)

For mean instead of median we have A047993.
For product instead of median we have A325039, ranks A325040.
For union instead of conjugate we have A360245, complement A360244.
Median of conjugate by rank is A363219.
These partitions are ranked by A363261.
A000700 counts self-conjugate partitions, ranks A088902.
A046682 and A352487-A352490 pertain to excedance set.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A330644 counts non-self-conjugate partitions (twice A000701), ranks A352486.
A352491 gives n minus Heinz number of conjugate.
Cf. A000975, A067538, A114638, A360068, A360242, A360248, A362617, A362618, A362621, A363223, A363260.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A329382 Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A118199 Number of partitions of n having no parts equal to the size of their Durfee squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A325045 Number of factorizations of n whose conjugate as an integer partition has no ones.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A349149 Number of even-length integer partitions of n with at most one odd part in the conjugate partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica