A277579 -id:A277579 - cp's OEIS frontend

A349151 Heinz numbers of integer partitions with alternating sum <= 1.

1, 2, 4, 6, 8, 9, 15, 16, 18, 24, 25, 32, 35, 36, 49, 50, 54, 60, 64, 72, 77, 81, 96, 98, 100, 121, 128, 135, 140, 143, 144, 150, 162, 169, 196, 200, 216, 221, 225, 240, 242, 256, 288, 289, 294, 308, 315, 323, 324, 338, 361, 375, 384, 392, 400, 437, 441, 450

Offset: 1

Gus Wiseman, Nov 10 2021

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 alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so these are also Heinz numbers of partitions with at most one odd conjugate part.

			The terms and their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   25: {3,3}
   32: {1,1,1,1,1}
   35: {3,4}
   36: {1,1,2,2}
   49: {4,4}

The case of alternating sum 0 is A000290.
These partitions are counted by A100824.
These are the positions of 0's and 1's in A344616.
The case of alternating sum 1 is A345958.
The conjugate partitions are ranked by A349150.
A000041 counts integer partitions.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A106529 ranks balanced partitions, counted by A047993.
A122111 is a representation of partition conjugation.
A257991 counts odd prime indices.
A316524 gives the alternating sum of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000070, A000700, A001222, A027187, A027193, A215366, A277103, A277579, A326841, A349149, A349158.

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[100],ats[Reverse[primeMS[#]]]<=1&]

Equals A000290 \/ A345958 decapitated.

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).

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]&]

cp's OEIS Frontend

A349151 Heinz numbers of integer partitions with alternating sum <= 1.

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

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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