proposed

approved

editing

proposed

The case of partitions ranked by Heinz numbers is A374706.

- Counts of six types of maximal runs: A124765, A124766, A124767, A124768, A124769, A333381.

- Leaders of six types of runs: A374251, A374515, A374629, A374683, A374740, A374757.

Six types of runs:

- Count: A124766, A124765, A124768, A124769, A333381, A124767.

- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.

- Rank: A375123, A375124, A375125, A375126, A375127, A373948.

Cf. A106356, A188920, A189076, A233564, A238343, A272919, A333213, A373949, `A374634, A374700, A374759, A374761, A374767.

Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.

Cf. A106356, A188920, A189076, A233564, A238343, A272919, A333213, A373949, `A374634, A374700, A374759, A374761, A374767.

allocated for Gus WisemanSum of leaders of strictly decreasing runs in the n-th composition in standard order.

0, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 5, 6, 5, 4, 5, 6, 3, 5, 5, 6, 5, 6, 5, 5, 4, 5, 5, 6, 5, 4, 5, 6, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 7, 6, 5, 6, 4, 4, 6, 6, 7, 6, 5, 4, 6, 5, 6, 6, 7, 6, 5, 6, 7, 6, 6

0,3

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

The maximal strictly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)) with leaders (3,2,2,2,5,1,1), so a(1234567) = 16.

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

Table[Total[First/@Split[stc[n], Greater]], {n, 0, 100}]

Row sums of A374757.

For leaders of constant runs we have A373953.

For leaders of anti-runs we have A374516.

For leaders of weakly increasing runs we have A374630.

For length instead of sum we have A124769.

The opposite version is A374684, sum of A374683 (length A124768).

The weak version is A374741, sum of A374740 (length A124765).

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1).

- Leader is A065120.

- Parts are listed by A066099.

- Number of adjacent equal pairs is A124762, unequal A333382.

- Counts of six types of maximal runs: A124765, A124766, A124767, A124768, A124769, A333381.

- Run-length transform is A333627, sum A070939.

- Run-compression transform is A373948.

- Ranks of contiguous compositions are A374249, counted by A274174.

- Leaders of six types of runs: A374251, A374515, A374629, A374683, A374740, A374757.

- Ranks of non-contiguous compositions are A374253, counted by A335548.

Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, `A374634, A374700, A374759, A374761, A374767.

allocated

nonn

Gus Wiseman, Jul 29 2024

approved

editing

allocated for Gus Wiseman

allocated

approved