A374689 Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.

1, 1, 1, 3, 3, 6, 10, 13, 21, 32, 48, 66, 101, 144, 207, 298, 415, 592, 833, 1163, 1615, 2247, 3088, 4259, 5845, 7977, 10862, 14752, 19969, 26941, 36310, 48725, 65279, 87228, 116274, 154660, 205305, 271879, 359400, 474157, 624257, 820450, 1076357, 1409598

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. The weakly decreasing version is A374697.

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.: Product_{i>0} (1 + (x^i)*Product_{j>i} (1 + x^j)). - John Tyler Rascoe, Jul 29 2024

Example

The a(0) = 1 through a(8) = 21 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)
                (12)  (13)  (14)   (15)   (16)   (17)
                (21)  (31)  (23)   (24)   (25)   (26)
                            (32)   (42)   (34)   (35)
                            (41)   (51)   (43)   (53)
                            (212)  (123)  (52)   (62)
                                   (213)  (61)   (71)
                                   (231)  (124)  (125)
                                   (312)  (214)  (134)
                                   (321)  (241)  (215)
                                          (313)  (251)
                                          (412)  (314)
                                          (421)  (323)
                                                 (341)
                                                 (413)
                                                 (431)
                                                 (512)
                                                 (521)
                                                 (2123)
                                                 (2312)
                                                 (3212)

Mathematica

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

Prog

(PARI)
C_x(N) = {my(x='x+O('x^N), h=prod(i=1, N, 1+(x^i)*prod(j=i+1, N, 1+x^j))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, Jul 29 2024

Crossrefs

The weak version appears to be A189076.

Ranked by positions of strictly decreasing rows in A374683.

The opposite version is A374762.

Types of runs (instead of strictly increasing):

- For leaders of identical runs we have A000041.

- For leaders of anti-runs we have A374680.

- For leaders of weakly increasing runs we have A188920.

- For leaders of weakly decreasing runs we have A374746.

- For leaders of strictly decreasing runs we have A374763.

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

- For identical leaders we have A374686, ranks A374685.

- For distinct leaders we have A374687, ranks A374698.

- For strictly increasing leaders we have A374688.

- 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.

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, A304969, A333213, A374632, A374635, A374640, A374679.

Sequence in context: A236345 A212991 A226642 * A266137 A265506 A366317

Adjacent sequences: A374686 A374687 A374688 * A374690 A374691 A374692

Keyword

nonn

Author

Gus Wiseman, Jul 27 2024

Extensions

a(26) onwards from John Tyler Rascoe, Jul 29 2024

Status

approved