cp's OEIS Frontend

A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

%I A344618 #9 Jun 08 2021 07:26:39
%S A344618 0,1,2,0,3,-1,1,1,4,-2,0,2,2,0,2,0,5,-3,-1,3,1,1,3,-1,3,-1,1,1,3,-1,1,
%T A344618 1,6,-4,-2,4,0,2,4,-2,2,0,2,0,4,-2,0,2,4,-2,0,2,2,0,2,0,4,-2,0,2,2,0,
%U A344618 2,0,7,-5,-3,5,-1,3,5,-3,1,1,3,-1,5,-3,-1,3
%N A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.
%C A344618 Up to sign, same as A124754.
%C A344618 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A344618 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.
%e A344618 The sequence of nonnegative integers together with the corresponding standard compositions and their reverse-alternating sums begins:
%e A344618   0:     () ->  0    15: (1111) ->  0    30:  (1112) ->  1
%e A344618   1:    (1) ->  1    16:    (5) ->  5    31: (11111) ->  1
%e A344618   2:    (2) ->  2    17:   (41) -> -3    32:     (6) ->  6
%e A344618   3:   (11) ->  0    18:   (32) -> -1    33:    (51) -> -4
%e A344618   4:    (3) ->  3    19:  (311) ->  3    34:    (42) -> -2
%e A344618   5:   (21) -> -1    20:   (23) ->  1    35:   (411) ->  4
%e A344618   6:   (12) ->  1    21:  (221) ->  1    36:    (33) ->  0
%e A344618   7:  (111) ->  1    22:  (212) ->  3    37:   (321) ->  2
%e A344618   8:    (4) ->  4    23: (2111) -> -1    38:   (312) ->  4
%e A344618   9:   (31) -> -2    24:   (14) ->  3    39:  (3111) -> -2
%e A344618   10:  (22) ->  0    25:  (131) -> -1    40:    (24) ->  2
%e A344618   11: (211) ->  2    26:  (122) ->  1    41:   (231) ->  0
%e A344618   12:  (13) ->  2    27: (1211) ->  1    42:   (222) ->  2
%e A344618   13: (121) ->  0    28:  (113) ->  3    43:  (2211) ->  0
%e A344618   14: (112) ->  2    29: (1121) -> -1    44:   (213) ->  4
%e A344618 Triangle begins (row lengths A011782):
%e A344618   0
%e A344618   1
%e A344618   2  0
%e A344618   3 -1  1  1
%e A344618   4 -2  0  2  2  0  2  0
%e A344618   5 -3 -1  3  1  1  3 -1  3 -1  1  1  3 -1  1  1
%t A344618 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344618 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]]
%t A344618 Table[sats[stc[n]],{n,0,100}]
%Y A344618 Up to sign, same as the reverse version A124754.
%Y A344618 The version for Heinz numbers of partitions is A344616.
%Y A344618 Positions of zeros are A344619.
%Y A344618 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344618 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344618 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344618 A116406 counts compositions with alternating sum >= 0.
%Y A344618 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344618 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A344618 All of the following pertain to compositions in standard order:
%Y A344618 - The length is A000120.
%Y A344618 - Converting to reversed ranking gives A059893.
%Y A344618 - The rows are A066099.
%Y A344618 - The sum is A070939.
%Y A344618 - The runs are counted by A124767.
%Y A344618 - The reversed version is A228351.
%Y A344618 - Strict compositions are ranked by A233564.
%Y A344618 - Constant compositions are ranked by A272919.
%Y A344618 - The Heinz number is A333219.
%Y A344618 - Anti-run compositions are ranked by A333489.
%Y A344618 Cf. A000070, A000097, A003242, A028260, A119899, A239830, A344605, A344607, A344608, A344650, A344739.
%K A344618 sign,tabf
%O A344618 0,3
%A A344618 _Gus Wiseman_, Jun 03 2021