|
%I #9 Oct 31 2022 15:23:45
%S 0,-1,1,0,0,-2,0,-1,0,2,1,-1,0,1,0,0,0,-3,-1,-2,0,1,0,-1,-1,1,-1,0,0,
%T 3,2,-2,1,0,1,-1,0,0,2,0,1,-1,0,0,1,0,0,0,0,-4,-2,-3,0,0,-1,-1,-2,1,
%U -2,0,0,2,1,-2,0,0,0,-1,0,-1,2,-1,1,-1,-1,0,1,-1
%N Triangle read by rows whose n-th row lists the first differences of the n-th composition in standard order (row n of A066099).
%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 Triangle begins (dots indicate empty rows):
%e 1: .
%e 2: .
%e 3: 0
%e 4: .
%e 5: -1
%e 6: 1
%e 7: 0 0
%e 8: .
%e 9: -2
%e 10: 0
%e 11: -1 0
%e 12: 2
%e 13: 1 -1
%e 14: 0 1
%e 15: 0 0 0
%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Table[Differences[stc[n]],{n,100}]
%Y See link for sequences related to standard compositions.
%Y First differences of rows of A066099.
%Y The version for Heinz numbers of partitions is A355536, ranked by A253566.
%Y The partial sums instead of first differences are A358134.
%Y Row sums are A358135.
%Y A011782 counts compositions.
%Y A351014 counts distinct runs in standard compositions.
%Y Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A242628, A357135, A357187.
%K sign,tabf
%O 3,6
%A _Gus Wiseman_, Oct 31 2022
|
