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proposed

stc[n_]:=Differences[Prepend[Join@@@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

The standardStandard compositions are listed by A066099.

Cf. A029837, A124767, `, A175413, A238279/A333755, `A274174, A333381, A353847, `, A353849.

A003242 counts anti-run compositions, ranked by A333489.

Cf. A003242 comps_antirun, A029837 stc_sum, A124767 stc_runs, A175413 binexp_has_dstnct_runs, A181819 pri_shadow, A238279 tri_comps_k_runs_wo_0, A274174 comps_runs_tgthr, A333381 stc_co_runs, A333489 stc_anti, A333755 tri_comps_runs, `A353835 prix_numdstnct_runsums, A353839 prix_wo_dstnct_runsums, A353847 stc_runsums, A353849 stc_numdstnct_runsums, `A353864 ptns_rucksack, `A353866 h_rucksack.

Cf. A029837, A124767, `A175413, A238279/A333755, `A274174, A333381, A353847, `A353849.

First rowcolumn is 1 followed by A000005.

allocatedNumber of ways to choose a divisor of each part of the n-th composition in standard order such that no adjacent fordivisors Gusare Wisemanequal.

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

0,3

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).

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)

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}]

First row is 1 followed by A000005.

Row-sums are A011782.

The standard compositions are listed by A066099.

Positions of 0's are A354904.

Positions of first appearances are A354905.

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. A003242 comps_antirun, A029837 stc_sum, A124767 stc_runs, A175413 binexp_has_dstnct_runs, A181819 pri_shadow, A238279 tri_comps_k_runs_wo_0, A274174 comps_runs_tgthr, A333381 stc_co_runs, A333489 stc_anti, A333755 tri_comps_runs, `A353835 prix_numdstnct_runsums, A353839 prix_wo_dstnct_runsums, A353847 stc_runsums, A353849 stc_numdstnct_runsums, `A353864 ptns_rucksack, `A353866 h_rucksack.

allocated

nonn,tabf

Gus Wiseman, Jun 11 2022

approved

editing

allocating

allocated

allocated for Gus Wiseman

allocating

approved