A374746 Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.

1, 1, 2, 3, 5, 7, 12, 18, 31, 51, 86, 143, 241, 397, 657, 1082, 1771, 2889, 4697, 7605, 12269, 19720, 31580, 50412, 80205, 127208, 201149, 317171, 498717, 782076, 1223230, 1908381, 2969950, 4610949, 7141972, 11037276, 17019617, 26188490, 40213388, 61624824

Offset

0,3

Comments

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

Links

Table of n, a(n) for n=0..39.

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

Example

The a(0) = 1 through a(7) = 18 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (312)     (331)
                                (11111)  (321)     (412)
                                         (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3112)
                                         (111111)  (3121)
                                                   (3211)
                                                   (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

Mathematica

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

Crossrefs

Ranked by positions of strictly decreasing rows in A374740, opp. A374629.

Types of runs (instead of weakly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we have A188920.

- For leaders of anti-runs we have A374680.

- For leaders of strictly increasing runs we have A374689.

- For leaders of strictly decreasing runs we have A374763.

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

- For weakly increasing leaders we appear to have A188900.

- For identical leaders we have A374742.

- For distinct leaders we have A374743, ranks A374701.

- For strictly increasing leaders we have A374745, opposite A374634.

- For weakly decreasing leaders we have A374747.

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.

A374748 counts compositions by sum of leaders of weakly decreasing runs.

Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.

Sequence in context: A137713 A326490 A191385 * A345669 A335093 A143642

Adjacent sequences: A374742 A374743 A374744 * A374747 A374748 A374750

Keyword

nonn,new

Author

Gus Wiseman, Jul 26 2024

Extensions

a(24)-a(39) from Alois P. Heinz, Jul 26 2024

Status

approved