A357623 - OEIS

%I #7 Oct 08 2022 09:39:55

%S 0,1,2,0,3,1,-1,-1,4,2,0,0,-2,-2,-2,0,5,3,1,1,-1,-1,-1,1,-3,-3,-3,-1,

%T -3,-1,1,1,6,4,2,2,0,0,0,2,-2,-2,-2,0,-2,0,2,2,-4,-4,-4,-2,-4,-2,0,0,

%U -4,-2,0,0,2,2,2,0,7,5,3,3,1,1,1,3,-1,-1,-1,1,-1

%N Skew-alternating sum of the n-th composition in standard order.

%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 358-th composition is (2,1,3,1,2) so a(358) = 2 - 1 - 3 + 1 + 2 = 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 Table[skats[stc[n]],{n,0,100}]

%Y See link for sequences related to standard compositions.

%Y Positions of positive firsts appear to be A029744.

%Y The half-alternating form is A357621, reverse A357622.

%Y The reverse version is A357624.

%Y Positions of zeros are A357627, reverse A357628.

%Y The version for prime indices is A357630.

%Y The version for Heinz numbers of partitions is A357634.

%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, A344619, A357136, A357182, A357183, A357184, A357185, A357625, A357626.

%K sign

%O 0,3

%A _Gus Wiseman_, Oct 08 2022