A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 7, 9, 6, 3, 1, 1, 11, 14, 12, 6, 3, 1, 1, 15, 23, 20, 12, 6, 3, 1, 1, 22, 34, 35, 21, 12, 6, 3, 1, 1, 30, 52, 56, 38, 21, 12, 6, 3, 1, 1, 42, 75, 91, 62, 38, 21, 12, 6, 3, 1, 1, 56, 109, 140, 103, 63, 38, 21, 12, 6, 3, 1, 1

Offset

0,4

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.

Also the number of reversed integer partitions of 2n with alternating sum 2k.

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
   1
   1   1
   2   1   1
   3   3   1   1
   5   5   3   1   1
   7   9   6   3   1   1
  11  14  12   6   3   1   1
  15  23  20  12   6   3   1   1
  22  34  35  21  12   6   3   1   1
  30  52  56  38  21  12   6   3   1   1
  42  75  91  62  38  21  12   6   3   1   1
  56 109 140 103  63  38  21  12   6   3   1   1
  77 153 215 163 106  63  38  21  12   6   3   1   1
Row n = 5 counts the following partitions:
  (55)          (442)        (433)      (622)    (811)  (10)
  (3322)        (541)        (532)      (721)
  (4411)        (22222)      (631)      (61111)
  (222211)      (32221)      (42211)
  (331111)      (33211)      (52111)
  (22111111)    (43111)      (4111111)
  (1111111111)  (2221111)
                (3211111)
                (211111111)

Mathematica

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], k==sats[#]&]], {n, 0, 15, 2}, {k, 0, n, 2}]

Crossrefs

The columns with initial 0's removed appear to converge to A006330.

The odd version is A239829.

The non-reversed version is A239830.

Row sums are A344611, odd bisection of A344607.

Including odd n and negative k gives A344612 (strict: A344739).

The strict case is A344649 (row sums: A344650).

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A103919 counts partitions by sum and alternating sum.

A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).

A316524 is the alternating sum of the prime indices of n (reverse: A344616).

A325534/A325535 count separable/inseparable partitions.

A344604 counts wiggly compositions with twins.

A344618 gives reverse-alternating sums of standard compositions.

Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.

Sequence in context: A157219 A167040 A054450 * A337009 A174802 A238346

Adjacent sequences: A344607 A344608 A344609 * A344611 A344612 A344613

Keyword

nonn,tabl

Author

Gus Wiseman, May 31 2021

Status

approved