A374764 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly increasing.

1, 1, 2, 4, 7, 13, 23, 40, 69, 118, 199, 333, 553, 911, 1492, 2428, 3928, 6323, 10129, 16151, 25646, 40560, 63905, 100332, 156995, 244877, 380803, 590479, 913100, 1408309, 2166671, 3325445, 5092283, 7780751, 11863546, 18052080, 27415291, 41556849, 62879053, 94975305, 143213145

Offset

0,3

Comments

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the maxima are weakly increasing [but weakly decreasing works too]. The strictly increasing version is A374762.

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).

Formula

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

Example

The composition (1,1,2,1) has strictly decreasing runs ((1),(1),(2,1)) with leaders (1,1,2) so is counted under a(5).
The composition (1,2,1,1) has strictly decreasing runs ((1),(2,1),(1)) with leaders (1,2,1) so is not counted under a(5).
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)
                        (121)   (113)
                        (1111)  (122)
                                (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (11111)

Mathematica

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

Prog

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

Crossrefs

For partitions instead of compositions we have A034296.

For strictly increasing leaders we have A374688.

The opposite version is A374697.

Other types of runs (instead of strictly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of anti-runs we have A374681.

- For leaders of weakly increasing runs we have A374635.

- For leaders of strictly increasing runs we have A374690.

- For leaders of weakly decreasing runs we have A188900.

Other types of run-leaders (instead of weakly increasing):

- For identical leaders we have A374760, ranks A374759.

- For distinct leaders we have A374761, ranks A374767.

- For strictly increasing leaders we have A374762.

- For weakly decreasing leaders we have A374765.

- For strictly decreasing leaders we have A374763.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

A335548 counts non-contiguous compositions, ranks A374253.

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

Cf. A106356, A188920, A238343, A261982, A333213, A374687, A374679, A374680, A374742, A374743, A374747.

Sequence in context: A128609 A335471 A168043 * A374632 A114832 A239553

Adjacent sequences: A374761 A374762 A374763 * A374765 A374766 A374767

Keyword

nonn

Author

Gus Wiseman, Jul 30 2024

Extensions

a(24) onwards from Andrew Howroyd, Jul 31 2024

Status

approved