A357627 - OEIS

%I #5 Oct 08 2022 14:16:11

%S 0,3,10,11,15,36,37,38,43,45,54,55,58,59,63,136,137,138,140,147,149,

%T 153,166,167,170,171,175,178,179,183,190,191,204,205,206,212,213,214,

%U 219,221,228,229,230,235,237,246,247,250,251,255,528,529,530,532,536

%N Numbers k such that the k-th composition in standard order has skew-alternating sum 0.

%C We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

%C 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.

%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     3: (1,1)

%e    10: (2,2)

%e    11: (2,1,1)

%e    15: (1,1,1,1)

%e    36: (3,3)

%e    37: (3,2,1)

%e    38: (3,1,2)

%e    43: (2,2,1,1)

%e    45: (2,1,2,1)

%e    54: (1,2,1,2)

%e    55: (1,2,1,1,1)

%e    58: (1,1,2,2)

%e    59: (1,1,2,1,1)

%e    63: (1,1,1,1,1,1)

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

%t skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];

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

%Y See link for sequences related to standard compositions.

%Y The alternating form is A344619.

%Y Positions of zeros in A357623.

%Y The half-alternating form is A357625, reverse A357626.

%Y The reverse version is A357628.

%Y The version for prime indices is A357632.

%Y The version for Heinz numbers of partitions is A357636.

%Y A124754 gives alternating sum of standard compositions, reverse A344618.

%Y A357637 counts partitions by half-alternating sum, skew A357638.

%Y A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.

%Y Cf. A001700, A001511, A053251, A357136, A357182-A357185, A357621, A357624, A357630, A357634, A357640.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 08 2022