A354578 Number of ways to choose a divisor of each part of the n-th composition in standard order such that no adjacent divisors are equal.

1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 0, 0, 2, 2, 3, 0, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 4, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 0, 1, 1, 0, 0, 1, 2, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 0, 5, 2, 2, 0, 5, 1, 3, 0, 1, 1, 0, 0, 3, 3, 5, 0, 3, 1, 1

Offset: 0

Author

Gus Wiseman, Jun 11 2022

Keywords

• nonn
• tabf

Comments

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). Then a(n) is the number of integer compositions whose run-sums constitute the n-th composition in standard order (graded reverse-lexicographic, A066099).

Examples

			The terms 2^(n - 1) through 2^n - 1 sum to 2^n. As a triangle:
  1
  1
  2 0
  2 1 1 0
  3 1 2 0 1 1 0 0
  2 2 3 0 3 1 1 0 2 1 1 0 0 0 0 0
The a(n) compositions for selected n:
  n=1: n=2:   n=8:       n=32:          n=68:        n=130:
----------------------------------------------------------------------
  (1)  (2)    (4)        (6)            (4,3)        (6,2)
       (1,1)  (2,2)      (3,3)          (2,2,3)      (3,3,2)
              (1,1,1,1)  (2,2,2)        (4,1,1,1)    (6,1,1)
                         (1,1,1,1,1,1)  (1,1,1,1,3)  (3,3,1,1)
                                        (2,2,1,1,1)  (2,2,2,1,1)
                                                     (1,1,1,1,1,1,2)
		

Crossrefs

First column is 1 followed by A000005.

Row-sums are A011782.

Standard compositions are listed by A066099.

Positions of 0's are A354904.

Positions of first appearances are A354905.

A003242 counts anti-run compositions, ranked by A333489.

A005811 counts runs in binary expansion.

A300273 ranks collapsible partitions, counted by A275870.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353851 counts compositions with all equal run-sums, ranked by A353848.

A353840-A353846 pertain to partition run-sum trajectory.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

A353853-A353859 pertain to composition run-sum trajectory.

A353860 counts collapsible compositions.

A353863 counts run-sum-complete partitions.

A354584 gives run-sums of prime indices, rows ranked by A353832.

Cf. A029837, A124767, A175413, A238279/A333755, A333381, A353847, A353849.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,30}]