A349155 Numbers k such that the k-th composition in standard order has sum equal to negative twice its reverse-alternating sum.

0, 9, 130, 135, 141, 153, 177, 193, 225, 2052, 2059, 2062, 2069, 2074, 2079, 2089, 2098, 2103, 2109, 2129, 2146, 2151, 2157, 2169, 2209, 2242, 2247, 2253, 2265, 2289, 2369, 2434, 2439, 2445, 2457, 2481, 2529, 2561, 2689, 2818, 2823, 2829, 2841, 2865, 2913

Offset: 1

Author

Gus Wiseman, Nov 22 2021

Keywords

• nonn

Comments

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.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Examples

			The terms and corresponding compositions begin:
     0: ()
     9: (3,1)
   130: (6,2)
   135: (5,1,1,1)
   141: (4,1,2,1)
   153: (3,1,3,1)
   177: (2,1,4,1)
   193: (1,6,1)
   225: (1,1,5,1)
  2052: (9,3)
  2059: (8,2,1,1)
  2062: (8,1,1,2)
  2069: (7,2,2,1)
  2074: (7,1,2,2)
  2079: (7,1,1,1,1,1)
  2089: (6,2,3,1)
  2098: (6,1,3,2)
  2103: (6,1,2,1,1,1)
		

Crossrefs

These compositions are counted by A224274 up to 0's.

An unordered version is A348617, counted by A001523 up to 0's.

The positive version is A349153, unreversed A348614.

The unreversed version is A349154.

Positive unordered unreversed: A349159, counted by A000712 up to 0's.

A positive unordered version is A349160, counted by A006330 up to 0's.

A003242 counts Carlitz compositions.

A011782 counts compositions.

A025047 counts alternating or wiggly compositions, complement A345192.

A034871, A097805, and A345197 count compositions by alternating sum.

A103919 counts partitions by alternating sum, reverse A344612.

A116406 counts compositions with alternating sum >=0, ranked by A345913.

A138364 counts compositions with alternating sum 0, ranked by A344619.

Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.

Statistics of standard compositions:

- The compositions themselves are the rows of A066099.

- Number of parts is given by A000120, distinct A334028.

- Sum and product of parts are given by A070939 and A124758.

- Maximum and minimum parts are given by A333766 and A333768.

- Heinz number is given by A333219.

Classes of standard compositions:

- Partitions and strict partitions are ranked by A114994 and A333256.

- Multisets and sets are ranked by A225620 and A333255.

- Strict and constant compositions are ranked by A233564 and A272919.

- Carlitz compositions are ranked by A333489, complement A348612.

- Alternating compositions are ranked by A345167, complement A345168.

Programs

Mathematica

stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==-2*sats[stc[#]]&]