A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

1, 1, 0, 0, 2, 3, 2, 5, 12, 22, 26, 58, 100, 203, 282, 616, 962, 2045, 2982, 6518, 9858, 21416, 31680, 69623, 104158, 228930, 339978, 751430, 1119668, 2478787, 3684082, 8182469, 12171900, 27082870, 40247978, 89748642, 133394708, 297933185, 442628598, 990210110

Offset: 0

Author

Gus Wiseman, Sep 28 2022

Keywords

• nonn

Comments

A composition of n is a finite 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(1) = 1 through a(8) = 12 compositions:
  (1)  (13)  (113)  (24)  (124)  (35)
       (31)  (212)  (42)  (151)  (53)
             (311)        (223)  (1115)
                          (322)  (1151)
                          (421)  (1214)
                                 (1313)
                                 (1412)
                                 (1511)
                                 (2141)
                                 (3131)
                                 (4121)
                                 (5111)
		

Crossrefs

For product instead of length we have A114220.

For sum equal to twice alternating sum we have A262977, ranked by A348614.

For product equal to sum we have A335405, ranked by A335404.

This is the absolute value version of A357182.

These compositions are ranked by A357185.

The case of partitions is A357189.

A003242 counts anti-run compositions, ranked by A333489.

A011782 counts compositions.

A025047 counts alternating compositions, ranked by A345167.

A124754 gives alternating sums of standard compositions.

A238279 counts compositions by sum and number of maximal runs.

A261983 counts non-anti-run compositions.

A357136 counts compositions by alternating sum.

Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.

Programs

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==Abs[ats[#]]&]],{n,0,15}]

Extensions

a(21)-a(39) from Alois P. Heinz, Sep 29 2022