A374697 Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.

1, 1, 2, 4, 8, 15, 29, 55, 103, 193, 360, 669, 1239, 2292, 4229, 7794, 14345, 26375, 48452, 88946, 163187, 299250, 548543, 1005172, 1841418, 3372603, 6175853, 11307358, 20699979, 37890704, 69351776, 126926194, 232283912, 425075191, 777848212, 1423342837, 2604427561

Offset: 0

Author

Gus Wiseman, Jul 27 2024

Keywords

• nonn

Comments

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

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are weakly decreasing [weakly increasing works too].

Examples

			The composition (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), so is not counted under a(12).
The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

The opposite version is A374764.

Ranked by positions of weakly decreasing rows in A374683.

Interchanging weak/strict appears to give A188920, opposite A358836.

Types of runs (instead of strictly increasing):

- For leaders of identical runs we have A000041.

- For leaders of anti-runs we have A374682.

- For leaders of weakly increasing runs we have A189076, complement A374636.

- For leaders of weakly decreasing runs we have A374747.

- For leaders of strictly decreasing runs we have A374765.

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

- For identical leaders we have A374686, ranks A374685.

- For distinct leaders we have A374687, ranks A374698.

- For weakly increasing leaders we have A374690.

- For strictly increasing leaders we have A374688.

- For strictly decreasing leaders we have A374689.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A335456 counts patterns matched by compositions.

A373949 counts compositions by run-compressed sum, opposite A373951.

A374700 counts compositions by sum of leaders of strictly increasing runs.

Cf. A000009, A106356, A238343, A261982, A333213, A374632, A374679, A374740.

Programs

Mathematica

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

PARI

seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))) \\ Andrew Howroyd, Jul 31 2024

Formula

G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j>=k+1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024

Extensions

a(26) onwards from Andrew Howroyd, Jul 31 2024