A045931 -id:A045931 - cp's OEIS frontend

A239832 Number of partitions of n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.

0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 3, 7, 6, 11, 11, 17, 19, 27, 31, 41, 51, 62, 79, 95, 121, 142, 182, 212, 269, 314, 393, 459, 570, 665, 816, 958, 1160, 1364, 1639, 1928, 2297, 2706, 3200, 3768, 4434, 5212, 6105, 7170, 8361, 9799, 11396, 13322, 15450, 18022

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of partitions of n having 1 more odd part than even, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are odd. Then c(n) = a(n+1) for n >= 0.

			The three partitions counted by a(10) are [10], [4,1,2,1,2], and [2,3,2,1,2].

Cf. A239833, A239835, A045931, A239871.

Column k=-1 of A240009.

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions *)
Table[Length[p[n]], {n, 0, 30}]  (* A239832 *)
(* Peter J. C. Moses, Mar 10 2014 *)

A239833 Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent and the first and last terms have the same parity.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 17, 22, 28, 36, 46, 58, 72, 92, 113, 141, 174, 216, 263, 324, 394, 481, 583, 707, 852, 1029, 1235, 1481, 1774, 2118, 2524, 3003, 3567, 4225, 5003, 5906, 6968, 8202, 9646, 11317, 13275, 15531, 18160, 21195, 24718, 28772

Offset: 0

Clark Kimberling, Mar 29 2014

			a(10) counts these 10 partitions:  [10], [1,8,1], [7,2,1], [3,6,1], [5,4,1], [5,3,2], [3,4,3], [4,1,2,1,2], [2,3,2,1,2], [1,2,1,2,1,2,1].

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

Cf. A239832, A239835, A045931, A239871.

b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, -1) +b(n$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

p[n_] := p[n] = Select[IntegerPartitions[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] == 1 &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions*)
t = Table[Length[p[n]], {n, 0, 60}] (* A239833 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]>n, 0, If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]]; a[n_] := b[n, n, -1] + b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

a(n) = A239832(n) + A239832(n+1) for n >= 0.

a(n) = A240009(n,-1) + A240009(n,1). - Alois P. Heinz, Apr 02 2014

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}]

A366749 Self-signed alternating sum of the prime indices of n.

Original entry on oeis.org

0, -1, 2, -2, -3, 1, 4, -3, 4, -4, -5, 0, 6, 3, -1, -4, -7, 3, 8, -5, 6, -6, -9, -1, -6, 5, 6, 2, 10, -2, -11, -5, -3, -8, 1, 2, 12, 7, 8, -6, -13, 5, 14, -7, 1, -10, -15, -2, 8, -7, -5, 4, 16, 5, -8, 1, 10, 9, -17, -3, 18, -12, 8, -6, 3, -4, -19, -9, -7, 0

Offset: 1

Gus Wiseman, Oct 23 2023

sign

We define the self-signed alternating sum of a multiset y to be Sum_{k in y} k*(-1)^k.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

With summands of 2^(n-1) we get A048675.
With summands of (-1)^k we get A195017.
The version for alternating prime indices is A346697 - A346698 = A316524.
Positions of zeros are A366748, counted by A239261.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A300061 lists numbers with even sum of prime indices, odd A300063.
A366528 adds up odd prime indices, counted by A113685.
A366531 adds up even prime indices, counted by A113686.
Cf. A000720, A019507, A045931, A055396, A061395, A325698, A325700, A366533.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
asum[y_]:=Sum[x*(-1)^x,{x,y}];
Table[asum[prix[n]],{n,100}]

a(n) = Sum_{k in A112798(n)} k*(-1)^k.

a(n) = A366531(n) - A366528(n).

A349159 Numbers whose sum of prime indices is twice their alternating sum.

Original entry on oeis.org

1, 12, 63, 66, 112, 190, 255, 325, 408, 434, 468, 609, 805, 832, 931, 946, 1160, 1242, 1353, 1380, 1534, 1539, 1900, 2035, 2067, 2208, 2296, 2387, 2414, 2736, 3055, 3108, 3154, 3330, 3417, 3509, 3913, 4185, 4340, 4503, 4646, 4650, 4664, 4864, 5185, 5684, 5863

Offset: 1

Gus Wiseman, Nov 23 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their alternating sum.

			The terms and their prime indices begin:
     1: ()
    12: (2,1,1)
    63: (4,2,2)
    66: (5,2,1)
   112: (4,1,1,1,1)
   190: (8,3,1)
   255: (7,3,2)
   325: (6,3,3)
   408: (7,2,1,1,1)
   434: (11,4,1)
   468: (6,2,2,1,1)
   609: (10,4,2)
   805: (9,4,3)
   832: (6,1,1,1,1,1,1)
   931: (8,4,4)
   946: (14,5,1)
  1160: (10,3,1,1,1)

These partitions are counted by A000712 up to 0's.
An ordered version is A348614, negative A349154.
The negative version is A348617.
The reverse version is A349160, counted by A006330 up to 0's.
A025047 counts alternating or wiggly compositions, complement A345192.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, and A345197 count compositions by alternating sum.
A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >= 0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000070, A000290, A001700, A028260, A045931, A120452, A195017, A241638, A257991, A257992, A325698, A345958, A349155.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==2*ats[primeMS[#]]&]

A056239(a(n)) = 2*A316524(a(n)).

A346697(a(n)) = 3*A346698(a(n)).

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}]

A239835 Number of partitions of n such that the absolute value of the difference between the number of odd parts and the number of even parts is <=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 15, 20, 26, 33, 44, 54, 71, 86, 113, 136, 175, 211, 268, 323, 403, 487, 601, 726, 885, 1068, 1292, 1556, 1867, 2244, 2678, 3208, 3809, 4547, 5379, 6398, 7542, 8937, 10506, 12404, 14542, 17110, 20011, 23465, 27381, 32006, 37267

Offset: 0

Clark Kimberling, Mar 29 2014

Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent, as in Example.

			a(8) counts these 8 partitions:  8, 161, 521, 341, 4121, 323, 3212, 21212.

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

Cf. A239832, A239833, A045931, A239871.

b:= proc(n, i, t) option remember; `if`(abs(t)-n>1, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

p[n_] := p[n] = Select[IntegerPartitions[n], Abs[Count[#, ?OddQ] - Count[#, ?EvenQ]] <= 1 &]; t = Table[p[n], {n, 0, 10}]
TableForm[t] (* shows the partitions *)
Table[Length[p[n]], {n, 0, 60}] (* A239835 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[Abs[t]-n>1, 0, If[n==0, 1, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

a(n) = A045931(n) + A239833(n) for n >= 0.

a(n) = Sum_{k=-1..1} A240009(n,k). - Alois P. Heinz, Apr 01 2014

A300789 Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 107, 108, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131

Offset: 1

Gus Wiseman, Mar 12 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

This sequence is conjectured to be the Heinz numbers of integer partitions in which the odd parts appear as many times in even as in odd positions.

			Sequence of integer partitions whose Young diagram can be tiled by dominos begins: (), (2), (11), (4), (22), (31), (211), (6), (1111), (8), (42), (51), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Wikipedia, Domino tiling

Cf. A000712, A000898, A001405, A004003, A045931, A097613, A099390, A299926, A300056, A300060, A300787, A300788, A304662.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while (l-> add(`if`(l[i]::odd,
       (-1)^i, 0), i=1..nops(l))<>0)(sort(map(i->
       numtheory[pi](i[1])$i[2], ifactors(k)[2]))) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[(-1)^Flatten[Position[primeMS[#],_?OddQ]]]===0&] (* Conjectured *)

A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 15, 21, 27, 38, 47, 63, 79, 106, 130, 170, 209, 272, 330, 422, 512, 653, 784, 986, 1183, 1482, 1765, 2191, 2604, 3218, 3804, 4666, 5504, 6726, 7898, 9592, 11240, 13602, 15880, 19122, 22277, 26733, 31048, 37102, 43003, 51232, 59220

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(7) = 8 partitions: (7), (511), (421), (331), (322), (31111), (22111), (1111111). Missing are: (61), (52), (43), (4111), (3211), (2221), (211111).

Even- and odd-indexed terms are A006330 and A001523 respectively, which add up to A000712.

Cf. A000898, A001405, A026010, A045931, A058696, A063886, A097613, A130780, A171966, A239241, A300788, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&]],{n,0,50}]

A325699 Number of distinct even prime indices of n minus the number of distinct odd prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 17 2019

sign

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

a(n) = A324967(n) - A324966(n).

Cf. A000712, A001221, A026010, A045931, A063886, A112798, A130780, A171966, A241638, A325698, A325700.

Table[Total[(-1)^PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]

G.f.: Sum_{k>=1} (-1)^k * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 12 2020

Additive with a(p^e) = (-1)^primepi(p). - Amiram Eldar, Jun 17 2024

cp's OEIS Frontend

A239832 Number of partitions of n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.

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A239833 Number of partitions of n having an ordering of parts in which no parts of equal parity are adjacent and the first and last terms have the same parity.

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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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A366749 Self-signed alternating sum of the prime indices of n.

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A349159 Numbers whose sum of prime indices is twice their alternating sum.

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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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A239835 Number of partitions of n such that the absolute value of the difference between the number of odd parts and the number of even parts is <=1.

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A300789 Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.

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A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.