%I #13 Sep 29 2022 12:57:02

%S 0,1,9,19,22,28,34,69,74,84,104,132,135,141,153,177,225,265,271,274,

%T 283,286,292,307,310,316,328,355,358,364,376,400,451,454,460,472,496,

%U 520,523,526,533,538,553,562,593,610,673,706,833,898,1041,1047,1053,1058

%N Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

%C A composition of n is a finite sequence of positive integers summing to n. 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.

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

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e The sequence together with the corresponding compositions begins:

%e 0: ()

%e 1: (1)

%e 9: (3,1)

%e 19: (3,1,1)

%e 22: (2,1,2)

%e 28: (1,1,3)

%e 34: (4,2)

%e 69: (4,2,1)

%e 74: (3,2,2)

%e 84: (2,2,3)

%e 104: (1,2,4)

%e 132: (5,3)

%e 135: (5,1,1,1)

%e 141: (4,1,2,1)

%e 153: (3,1,3,1)

%e 177: (2,1,4,1)

%e 225: (1,1,5,1)

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]

%Y See link for sequences related to standard compositions.

%Y For product equal to sum we have A335404, counted by A335405.

%Y For sum equal to twice alternating sum we have A348614, counted by A262977.

%Y These compositions are counted by A357182.

%Y For absolute value we have A357184, counted by A357183.

%Y The case of partitions is counted by A357189.

%Y A003242 counts anti-run compositions, ranked by A333489.

%Y A011782 counts compositions.

%Y A025047 counts alternating compositions, ranked by A345167.

%Y A032020 counts strict compositions, ranked by A233564.

%Y A124754 gives alternating sums of standard compositions.

%Y A238279 counts compositions by sum and number of maximal runs.

%Y A357136 counts compositions by alternating sum.

%Y Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

%K nonn

%O 1,3

%A _Gus Wiseman_, Sep 28 2022