|
%I #7 Oct 31 2022 15:23:26
%S 0,0,0,0,-1,1,0,0,-2,0,-1,2,0,1,0,0,-3,-1,-2,1,-1,0,-1,3,0,1,0,2,0,1,
%T 0,0,-4,-2,-3,0,-2,-1,-2,2,-1,0,-1,1,-1,0,-1,4,0,1,0,2,0,1,0,3,0,1,0,
%U 2,0,1,0,0,-5,-3,-4,-1,-3,-2,-3,1,-2,-1,-2,0,-2
%N Difference of first and last parts of the n-th composition in standard order.
%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>
%F a(n) = A001511(n) - A065120(n).
%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Table[-First[stc[n]]+Last[stc[n]],{n,1,100}]
%Y See link for sequences related to standard compositions.
%Y The first and last parts are A065120 and A001511.
%Y This is the first minus last part of row n of A066099.
%Y The version for Heinz numbers of partitions is A243055.
%Y Row sums of A358133.
%Y The partial sums of standard compositions are A358134, adjusted A242628.
%Y A011782 counts compositions.
%Y A333766 and A333768 give max and min in standard compositions, diff A358138.
%Y A351014 counts distinct runs in standard compositions.
%Y Cf. A000120, A001511, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187.
%K sign
%O 1,9
%A _Gus Wiseman_, Oct 31 2022
|
