A098123 -id:A098123 - cp's OEIS frontend

A242498 Number T(n,k) of compositions of n, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows.

1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 1, 0, 3, 2, 0, 1, 3, 4, 1, 4, 3, 0, 1, 1, 2, 1, 6, 9, 3, 5, 4, 0, 1, 4, 9, 6, 11, 16, 6, 6, 5, 0, 1, 1, 3, 3, 11, 24, 18, 19, 25, 10, 7, 6, 0, 1, 5, 16, 18, 28, 51, 40, 31, 36, 15, 8, 7, 0, 1, 1, 4, 6, 19, 51, 60, 65, 95, 75, 48, 49, 21, 9, 8, 0, 1

Offset: 0

Alois P. Heinz, May 16 2014

T(n,k) = T(n+k,-k).

			Triangle T(n,k) begins:
: n\k : -5 -4 -3  -2  -1   0   1   2   3   4   5   6  7  8  9 10 ...
+-----+---------------------------------------------------------
:  0  :                    1;
:  1  :                        1;
:  2  :                1,  0,  0,  1;
:  3  :                    2,  1,  0,  1;
:  4  :            1,  1,  0,  3,  2,  0,  1;
:  5  :                3,  4,  1,  4,  3,  0,  1;
:  6  :        1,  2,  1,  6,  9,  3,  5,  4,  0,  1;
:  7  :            4,  9,  6, 11, 16,  6,  6,  5,  0, 1;
:  8  :     1, 3,  3, 11, 24, 18, 19, 25, 10,  7,  6, 0, 1;
:  9  :        5, 16, 18, 28, 51, 40, 31, 36, 15,  8, 7, 0, 1;
: 10  :  1, 4, 6, 19, 51, 60, 65, 95, 75, 48, 49, 21, 9, 8, 0, 1;

Alois P. Heinz, Rows n = 0..120, flattened

Columns k=0-10 gives: A098123, A242499, A242500, A242501, A242502, A242503, A242504, A242505, A242506, A242507, A242508.
Row sums give A011782.
Diagonals include: A000012, A000004, A001477, A000217, A000290, A180415.
Row lengths give A016777(floor(n/2)).
Cf. A240009, A240021.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, expand(
      add(x^(j*(2*irem(i, 2)-1))*b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..20);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i<1, 0, Expand[Sum[x^(j*(2*Mod[i, 2]-1))*b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]] ; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

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}]

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}]

A242821 Number of compositions of n with equal number of even and odd parts, both counted without multiplicity.

Original entry on oeis.org

1, 0, 0, 2, 3, 11, 14, 33, 43, 82, 141, 260, 514, 1049, 2321, 4789, 10454, 21735, 46213, 94782, 196060, 398662, 810178, 1631089, 3278372, 6556096, 13088339, 26063238, 51824061, 102926784, 204239173, 405087125, 803109813, 1592179369, 3156298826, 6258390137

Offset: 0

Alois P. Heinz, May 23 2014

nonn

			a(3) = 2: [1,2], [2,1].
a(4) = 3: [1,1,2], [1,2,1], [2,1,1].
a(5) = 11: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [2,3], [3,2], [1,4], [4,1].
a(6) = 14: [1,1,1,1,2], [1,1,1,2,1], [1,1,2,1,1], [1,2,1,1,1], [2,1,1,1,1], [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [1,1,4], [1,4,1], [4,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=0 of A242626.

Cf. A098123 (counted with multiplicity).

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

A385604 Number of compositions of n such that the odd parts are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 25, 48, 86, 162, 292, 541, 978, 1794, 3247, 5919, 10712, 19451, 35184, 63729, 115199, 208327, 376333, 679842, 1227403, 2215695, 3998408, 7214274, 13014001, 23472678, 42331028, 76330880, 137627168, 248122171, 447301570, 806312371, 1453405651

Offset: 0

John Tyler Rascoe, Aug 02 2025

			a(5) = 14 counts all compositions of n = 5 except (1,3,1) and (3,1,1) since the odd parts are not weakly increasing.
The composition of n = 13 (2,1,1,4,2,3) has odd parts (1,1,3), so it is counted under a(13) = 1794.

Cf. A000009, A011782, A032021, A098123, A289249.

A_x(N) = {my(x='x+O('x^(N+1))); Vec((1-x^2)/(1-2*x^2)/prod(i=0,N, 1-x^(2*i+1)*(1-x^2)/(1-2*x^2)))}

G.f.: (1 - x^2)/( (1 - 2*x^2) * Product_{i>=0} (1 - x^(2*i + 1) * (1 - x^2)/(1 - 2*x^2)) ).

cp's OEIS Frontend

A242498 Number T(n,k) of compositions of n, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows.

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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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Mathematica

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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Mathematica

A242821 Number of compositions of n with equal number of even and odd parts, both counted without multiplicity.

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A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

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Mathematica

A385604 Number of compositions of n such that the odd parts are weakly increasing.

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