A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.

1, 1, 2, 3, 6, 10, 20, 36, 69, 130, 247, 467, 890, 1689, 3213, 6110, 11627, 22121, 42101, 80124, 152512, 290300, 552609, 1051953, 2002583, 3812326, 7257679, 13816867, 26304254, 50077792, 95338234, 181505938, 345554234, 657874081, 1252478707, 2384507463, 4539705261

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 (1,3,3,2,4,2) has weakly increasing runs ((1,3,3),(2,4),(2)), with leaders (1,2,2), so is counted under a(15).
The a(0) = 1 through a(6) = 20 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (122)    (114)
                        (1111)  (131)    (123)
                                (1112)   (132)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1122)
                                         (1131)
                                         (1212)
                                         (1221)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)
		

Links

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

Crossrefs

Ranked by positions of weakly increasing rows in A374629 (sums A374630).

Types of runs (instead of weakly increasing):

- For leaders of constant runs we have A000041.

- For leaders of weakly decreasing runs we have A188900.

- For leaders of anti-runs we have A374681.

- For leaders of strictly increasing runs we have A374690.

- For leaders of strictly decreasing runs we have A374764.

Types of run-leaders (instead of weakly increasing):

- 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 distinct leaders we have A374632, ranks A374768.

- For strictly increasing leaders we have A374634.

A003242 counts anti-run compositions.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

A335456 counts patterns matched by compositions.

A335548 counts non-contiguous compositions, ranks A374253.

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

Cf. A106356, A124766, A238343, A261982, A333213, A374518, A374687, A374761.

Programs

Mathematica

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

PARI

dfs(m, r, u) = 1 + sum(s=u, min(m, r-1), x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, s)*x^(s+t)/prod(i=s, t, 1-x^i)));
lista(nn) = Vec(dfs(nn, nn+1, 1) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

Extensions

More terms from Jinyuan Wang, Feb 13 2025