A116482 -id:A116482 - cp's OEIS frontend

A066898 Total number of even parts in all partitions of n.

0, 1, 1, 4, 5, 11, 15, 28, 38, 62, 85, 131, 177, 258, 346, 489, 648, 890, 1168, 1572, 2042, 2699, 3475, 4532, 5783, 7446, 9430, 12017, 15106, 19073, 23815, 29827, 37011, 46012, 56765, 70116, 86033, 105627, 128962, 157476, 191359, 232499, 281286, 340180, 409871

Offset: 1

Naohiro Nomoto, Jan 24 2002

Also sum of all even-indexed parts minus the sum of all odd-indexed parts, except the largest parts, of all partitions of n (cf. A206563). - Omar E. Pol, Feb 14 2012
From Omar E. Pol, Apr 06 2023: (Start)
Convolution of A000041 and A183063.
Convolution of A002865 and A362059.
a(n) is also the total number of even divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned even divisors are also all even parts of all partitions of n. (End)

			a(5) = 5 because in all the partitions of 5, namely [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], we have a total of 0+1+1+0+2+1+0=5 even parts.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

Cf. A000005, A000041, A002865, A006128, A066897, A116482, A183063, A206563, A209423, A362059.

Column 2 of A206563.

a066898 = p 0 1 where
   p e _             0 = e
   p e k m | m < k     = 0
           | otherwise = p (e + 1 - mod k 2) k (m - k) + p e (k + 1) m
-- Reinhard Zumkeller, Mar 09 2012

a066898 = length . filter even . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

g:=sum(x^(2*j)/(1-x^(2*j)),j=1..60)/product((1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50); # Emeric Deutsch, Feb 17 2006
A066898 := proc(n)
    add(numtheory[tau](k)*combinat[numbpart](n-2*k),k=1..n/2) ;
end proc: # R. J. Mathar, Jun 18 2016

f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
o[n_] := Sum[f[n, i], {i, 1, n, 2}]
e[n_] := Sum[f[n, i], {i, 2, n, 2}]
Table[o[n], {n, 1, 45}]  (* A066897 *)
Table[e[n], {n, 1, 45}]  (* A066898 *)
%% - %                   (* A209423 *)
(* Clark Kimberling, Mar 08 2012 *)
a[n_] := Sum[DivisorSigma[0, k] PartitionsP[n - 2k], {k, 1, n/2}]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 31 2016, after Vladeta Jovovic *)

a(n) = Sum_{k=1..floor(n/2)} tau(k)*numbpart(n-2*k). - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k=0..floor(n/2)} k*A116482(n,k). - Emeric Deutsch, Feb 17 2006
G.f.: (Sum_{j>=1} x^(2*j)/(1-x^(2*j)))/(Product_{j>=1} (1-x^j)). - Emeric Deutsch, Feb 17 2006
a(n) = A066897(n) - A209423(n) = A006128(n) - A066897(n). - Reinhard Zumkeller, Mar 09 2012
a(n) = (A006128(n) - A209423(n))/2. - Vaclav Kotesovec, May 25 2018
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(3*n/(2*Pi^2))) / (8*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

More terms from Vladeta Jovovic, Jan 26 2002

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)).

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}]

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.

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}]

A096778 Number of partitions of n with at most two even parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 45, 58, 75, 95, 121, 151, 189, 234, 289, 354, 433, 526, 637, 768, 923, 1105, 1319, 1569, 1861, 2202, 2597, 3056, 3587, 4201, 4908, 5723, 6658, 7732, 8961, 10367, 11971, 13802, 15884, 18253, 20942, 23992, 27445, 31353

Offset: 0

Vladeta Jovovic, Aug 16 2004

Also number of partitions of n+4 with exactly two even parts. Example: a(3)=3 because the partitions of 7 with exactly two even parts are [4,2,1], [3,2,2] and [2,2,1,1,1]. a(n)=A116482(n+4,2). - Emeric Deutsch, Feb 21 2006

			a(3)=3 because we have [3],[2,1] and [1,1,1].

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=2. - N. J. A. Sloane, Aug 31 2014

Cf. A038348.

Cf. A116482.

CoefficientList[ Series[(1/((1 - x^2)*(1 - x^4)))/Product[1 - x^(2i + 1), {i, 0, 50}], {x, 0, 48}], x] (* Robert G. Wilson v, Aug 16 2004 *)

G.f.: (1/((1-x^2)*(1-x^4)))/Product(1-x^(2*i+1), i=0..infinity). More generally, g.f. for number of partitions of n with at most k even parts is (1/Product(1-x^(2*i), i=1..k))/Product(1-x^(2*i+1), i=0..infinity).

a(n) ~ 3^(3/4) * n^(1/4) * exp(Pi*sqrt(n/3)) / (8*Pi^2). - Vaclav Kotesovec, May 29 2018

More terms from Robert G. Wilson v, Aug 17 2004

More terms from Emeric Deutsch, Feb 21 2006

cp's OEIS Frontend

A066898 Total number of even parts in all partitions of n.

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

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A350948 Number of integer partitions of n with as many even parts as even conjugate 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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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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A096778 Number of partitions of n with at most two even parts.

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