A357189 Number of integer partitions of n with the same length as alternating sum.

1, 1, 0, 0, 1, 1, 1, 2, 2, 4, 3, 5, 6, 9, 9, 13, 16, 23, 23, 34, 37, 54, 54, 78, 84, 120, 121, 170, 182, 252, 260, 358, 379, 517, 535, 725, 764, 1030, 1064, 1427, 1494, 1992, 2059, 2733, 2848, 3759, 3887, 5106, 5311, 6946, 7177, 9345, 9701, 12577, 12996, 16788

Offset: 0

Author

Gus Wiseman, Sep 30 2022

Keywords

• nonn

Comments

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The a(4) = 1 through a(13) = 9 partitions:
  31   311   42   322   53     333     64     443     75       553
                  421   5111   432     5221   542     5331     652
                               531     6211   641     6222     751
                               51111          52211   6321     52222
                                              62111   7311     53311
                                                      711111   62221
                                                               63211
                                                               73111
                                                               7111111
		

Crossrefs

For product equal to sum we have A001055, compositions A335405.

For product instead of length we have A004526, compositions A114220.

The version for compositions is A357182, ranked by A357184.

For sum equal to twice alternating sum we have A357189 (this sequence).

These partitions are ranked by A357486.

The reverse version is A357487, ranked by A357485.

A000041 counts partitions, strict A000009.

A025047 counts alternating compositions.

A103919 counts partitions by alternating sum, full triangle A344651.

A357136 counts compositions by alternating sum, full triangle A097805.

Cf. A051159, A070939, A131044, A262046, A262977, A301987, A357183.

Programs

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,0,30}]