A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

1, 1, 2, 4, 7, 13, 23, 40, 69, 119, 200, 335, 557, 917, 1499, 2433, 3920, 6280, 10004, 15837, 24946, 39087, 60952, 94606, 146203, 224957, 344748, 526239, 800251, 1212527, 1830820, 2754993, 4132192, 6178290, 9209308, 13686754, 20282733, 29973869, 44175908, 64936361

Offset: 0

Author

Gus Wiseman, Jul 23 2024

Keywords

• nonn

Comments

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

Examples

			The composition (4,2,2,1,1,3) has weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so is counted under a(13).
The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (211)   (113)
                        (1111)  (122)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)
		

Links

• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Ranked by A374768 = positions of distinct rows in A374629 (sums A374630).

Types of runs (instead of weakly increasing):

- For leaders of constant runs we have A274174, ranks A374249.

- For leaders of anti-runs we have A374518, ranks A374638.

- For leaders of strictly increasing runs we have A374687, ranks A374698.

- For leaders of weakly decreasing runs we have A374743, ranks A335467.

- For leaders of strictly decreasing runs we have A374761, ranks A374767.

Types of run-leaders (instead of distinct):

- For strictly decreasing leaders we appear to have A188920.

- For weakly decreasing leaders we appear to have A189076.

- For identical leaders we have A374631.

- For weakly increasing leaders we have A374635.

- For strictly increasing leaders we have A374634.

A003242 counts anti-run compositions.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

A335456 counts patterns matched by compositions.

A374637 counts compositions by sum of leaders of weakly increasing runs.

Cf. A106356, A124766, A238343, A261982, A333213, A335548, A373949, A373953.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,LessEqual]&]],{n,0,15}]

PARI

dfs(m, r, v) = 1 + sum(s=1, min(m, r-1), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)/prod(i=s, t, 1-x^i))));
lista(nn) = Vec(dfs(nn, nn+1, []) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

Extensions

More terms from Jinyuan Wang, Feb 13 2025