A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Author

Gus Wiseman, Nov 21 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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)
		

Crossrefs

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

Except for 0, a subset of A345919.

The positive version is A348614, reverse A349153.

An unordered version is A348617, counted by A001523.

The reverse version is A349155.

A positive unordered version is A349159, counted by A000712 up to 0's.

A000346 = even-length compositions with alt sum != 0, complement A001700.

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 sum and 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, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.

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.

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.

- Necklaces are ranked by A065609, dual A333764, reversed A333943.

- Alternating compositions are ranked by A345167, complement A345168.

Programs

Mathematica

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