A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.

1, 1, 1, 2, 2, 4, 5, 7, 11, 16, 21, 31, 45, 63, 87, 122, 170, 238, 328, 449, 616, 844, 1151, 1565, 2121, 2861, 3855, 5183, 6953, 9299, 12407, 16513, 21935, 29078, 38468, 50793, 66935, 88037, 115577, 151473, 198175, 258852, 337560, 439507, 571355, 741631

Offset

0,4

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 strictly decreasing.

Links

Christian Sievers, Table of n, a(n) for n = 0..500

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

Example

The a(0) = 1 through a(9) = 16 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
                (12)  (13)  (14)   (15)   (16)   (17)    (18)
                            (23)   (24)   (25)   (26)    (27)
                            (122)  (123)  (34)   (35)    (36)
                                   (132)  (124)  (125)   (45)
                                          (133)  (134)   (126)
                                          (142)  (143)   (135)
                                                 (152)   (144)
                                                 (233)   (153)
                                                 (1223)  (162)
                                                 (1232)  (234)
                                                         (243)
                                                         (1224)
                                                         (1233)
                                                         (1242)
                                                         (1323)

Mathematica

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

Crossrefs

The weak version is A374635.

Ranked by positions of strictly increasing rows in A374683 (sums A374684).

The opposite version is A374763.

Types of runs (instead of strictly increasing):

- For leaders of identical runs we have A000041.

- For leaders of anti-runs we have A374679.

- For leaders of weakly increasing runs we have A374634.

- For leaders of weakly decreasing runs we have A374745.

- For leaders of strictly decreasing runs we have A374762.

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

- For identical leaders we have A374686, ranks A374685.

- For distinct leaders we have A374687, ranks A374698.

- For strictly decreasing leaders we have A374689.

- For weakly increasing leaders we have A374690.

- For weakly decreasing leaders we have A374697.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

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

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

Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632.

Sequence in context: A359388 A325550 A362608 * A325330 A319160 A292382

Adjacent sequences: A374685 A374686 A374687 * A374689 A374690 A374691

Keyword

nonn,changed

Author

Gus Wiseman, Jul 27 2024

Extensions

a(26) and beyond from Christian Sievers, Aug 08 2024

Status

approved