A277103 -id:A277103 - cp's OEIS frontend

A277579 Number of partitions of n for which the number of even parts is equal to the positive alternating sum of the parts.

1, 0, 1, 1, 1, 2, 3, 3, 4, 6, 7, 9, 13, 15, 19, 25, 31, 38, 48, 59, 74, 90, 111, 136, 166, 201, 246, 297, 357, 431, 522, 621, 745, 892, 1063, 1263, 1503, 1780, 2109, 2491, 2941, 3463, 4077, 4783, 5616, 6576, 7689, 8981, 10486, 12207, 14209, 16516, 19178, 22231

Offset: 0

Emeric Deutsch and Alois P. Heinz, Oct 20 2016

nonn

In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, EP gives the number of even terms of a finite sequence of positive integers.
For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
Also the number of integer partitions of n with as many even parts as odd parts in the conjugate partition. - Gus Wiseman, Jul 26 2021

			a(9) = 6: [2,1,1,1,1,1,1,1], [3,2,1,1,1,1], [3,3,2,1], [4,2,2,1], [4,3,1,1], [5,4].
a(10) = 7: [1,1,1,1,1,1,1,1,1,1], [3,2,2,1,1,1], [3,3,1,1,1,1], [4,2,1,1,1,1], [4,3,2,1], [5,5], [6,4].
a(11) = 9: [2,1,1,1,1,1,1,1,1,1], [3,2,1,1,1,1,1,1], [3,3,2,1,1,1], [3,3,3,2], [4,2,2,1,1,1], [4,3,1,1,1,1], [5,2,2,2], [5,4,1,1], [6,5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The sign-sensitive version is A035457 (aerated version of A000009).
Comparing odd parts to odd conjugate parts gives A277103.
Comparing product of parts to product of conjugate parts gives A325039.
Comparing the rev-alt sum to that of the conjugate gives A345196.
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).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A000700, A006330, A027187, A027193, A236559, A257991, A325534, A325535, A344607, A344651.

with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 then ct := ct+1 else  end if end do: ct end proc: a := proc (n) local P, c, k: P := partition(n): c := 0: for k to nops(P) do if AS(P[k]) = EP(P[k]) then c := c+1 else  end if end do: c end proc: seq(a(n), n = 0 .. 30);
n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: EP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 0 then ct := ct+1 else  end if end do: ct end proc: P := partition(n): C := {}: for k to nops(P) do if AS(P[k]) = EP(P[k]) then C := `union`(C, {P[k]}) else  end if end do: C;
# alternative Maple program:
b:= proc(n, i, s, t) option remember; `if`(n=0,
      `if`(s=0, 1, 0), `if`(i<1, 0, b(n, i-1, s, t)+
      `if`(i>n, 0, b(n-i, i, s+t*i-irem(i+1, 2), -t))))
    end:
a:= n-> b(n$2, 0, 1):
seq(a(n), n=0..60);

b[n_, i_, s_, t_] := b[n, i, s, t] = If[n == 0, If[s == 0, 1, 0], If[i<1, 0, b[n, i-1, s, t] + If[i>n, 0, b[n-i, i, s + t*i - Mod[i+1, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)
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]&]],{n,0,15}] (* Gus Wiseman, Jul 26 2021 *)

def a(n):
    AS = lambda s: abs(sum((-1)^i*t for i,t in enumerate(s)))
    EP = lambda s: sum((t+1)%2 for t in s)
    return sum(AS(p) == EP(p) for p in Partitions(n))
print([a(n) for n in (0..30)]) # Peter Luschny, Oct 21 2016

A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

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

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), so a(n) counts even prime indices of n.

Positions of first appearances are A001248.
The triangular version is A116482.
Positions of zeros are A346635.
Subtracting from the number of odd conjugate parts gives A350941.
Subtracting from the number of odd parts gives A350942.
Subtracting from the number of even parts gives A350950.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847 (this sequence).
There are six 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.
- 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.
A122111 represents partition conjugation using Heinz numbers.
Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, 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]}]];
Table[Count[conj[primeMS[n]],_?EvenQ],{n,100}]

a(n) = A344616(n) - A350941(n).

a(n) = A257992(A122111(n)).

A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 6, 18, 21, 24, 54, 65, 70, 72, 84, 96, 133, 147, 162, 182, 189, 210, 216, 260, 280, 288, 319, 336, 384, 418, 429, 481, 486, 490, 525, 532, 546, 585, 588, 630, 648, 728, 731, 741, 754, 756, 840, 845, 864, 1007, 1029, 1040, 1120, 1152, 1197, 1254, 1258, 1276

Offset: 1

Gus Wiseman, Jan 27 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)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  54: (2,2,2,1)
  65: (6,3)
  70: (4,3,1)
  72: (2,2,1,1,1)
  84: (4,2,1,1)
  96: (2,1,1,1,1,1)

These partitions are counted by A045931.
The conjugate strict version is counted by A239241.
The conjugate version is A325698.
These are the positions of 0's in A350941.
Adding the conjugate condition gives A350946, all four equal A350947.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A325698: # of even parts = # of odd parts.
A349157: # of even parts = # of odd conjugate parts, counted by A277579.
A350848: # even conjugate parts = # odd conjugate 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.
A350945: # of even parts = # of even conjugate parts, counted by A350948.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A316524 = alternating sum of prime indices, reverse A344616.
Cf. A024619, A026424, A028260, A103919, A130780, A171966, A195017, A241638, A325700, A350849, A350942, A350949.

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

A344616(a(n)) = A350847(a(n)).

A257991(A122111(a(n))) = A257992(A122111(a(n))).

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 28 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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

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

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

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

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

Original entry on oeis.org

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)).

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 14 2022

nonn

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

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

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

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)).

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 3, -2, 1, 1, 2, 0, 0, -1, 3, 1, 0, 0, 3, -2, 1, 1, 2, -1, 0, 0, 2, 0, 1, 1, 5, -1, 1, -2, 0, 0, 0, -2, 3, 1, 0, 0, 3, 1, 1, 1, 4, -4, 1, -1, 2, 0, 0, -1, 2, -2, 0, 1, 1, 0, 1, 0, 5, -2, 1, 1, 3, -1, 0, 0, 2, 1, 0, 1, 2, -3, 0, 0, 5, -2, 1

Offset: 1

Gus Wiseman, Jan 28 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.

			First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
  192: (2,1,1,1,1,1,1)
   32: (1,1,1,1,1)
   48: (2,1,1,1,1)
    8: (1,1,1)
   12: (2,1,1)
    2: (1)
    1: ()
   15: (3,2)
    9: (2,2)
   77: (5,4)
   49: (4,4)
  221: (7,6)
  169: (6,6)

The conjugate version is A350849.
This is a hybrid of A195017 and A350941.
Positions of 0's are A350943.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents 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:
  A325698: # of even parts = # of odd parts.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # 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.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, 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]}]];
Table[Count[primeMS[n],?OddQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.

Original entry on oeis.org

1, 6, 84, 210, 490, 525, 2184, 2340, 5460, 9464, 12012, 12740, 12870, 13650, 14625, 19152, 22308, 30030, 34125, 43940, 45144, 55770, 59150, 66066, 70070, 70785, 75075, 79625, 82992, 88920

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)
    210: (4,3,2,1)
    490: (4,4,3,1)
    525: (4,3,3,2)
   2184: (6,4,2,1,1,1)
   2340: (6,3,2,2,1,1)
   5460: (6,4,3,2,1,1)
   9464: (6,6,4,1,1,1)
  12012: (6,5,4,2,1,1)
  12740: (6,4,4,3,1,1)
  12870: (6,5,3,2,2,1)
  13650: (6,4,3,3,2,1)
  14625: (6,3,3,3,2,2)
  19152: (8,4,2,2,1,1,1,1)
For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 is in the sequence.

These partitions are counted by A351978.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are six 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.
- 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.
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.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A239241, A241638, A325700, 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[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(a(n)) = A350847(a(n)) = A344616(a(n)).

A350949 Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.

Original entry on oeis.org

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 294, 315, 350, 416, 441, 490, 525, 624, 660, 735, 924, 990, 1088, 1100, 1386, 1540, 1560, 1632, 1650, 1715, 2184, 2310, 2340, 2401, 2432, 2600, 3267, 3276, 3388, 3640, 3648, 3900, 4080, 4125, 5082, 5324, 5390

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: ()
     2: (1)
     6: (2,1)
     9: (2,2)
    20: (3,1,1)
    30: (3,2,1)
    56: (4,1,1,1)
    75: (3,3,2)
    84: (4,2,1,1)
   125: (3,3,3)
   176: (5,1,1,1,1)
   210: (4,3,2,1)
   264: (5,2,1,1,1)
   294: (4,4,2,1)
   315: (4,3,2,2)
   350: (4,3,3,1)
   416: (6,1,1,1,1,1)

The second condition alone is A350944, counted by A277103.
The first condition alone is A350945, counted by A350948.
The case of all four statistics equal is A350947, counted by A351978.
These partitions are counted by A351976.
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:
- A350946, counted by A351977.
- A351980, counted by A351981.
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.
Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, 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[#],?OddQ]==Count[conj[primeMS[#]],?OddQ]&&Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?EvenQ]&]

Intersection of A350944 and A350945.
A257991(a(n)) = A344616(a(n)).
A257992(a(n)) = A350847(a(n)).
Closed under A122111 (conjugation).

cp's OEIS Frontend

A277579 Number of partitions of n for which the number of even parts is equal to the positive alternating sum of the parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.