A374762 Number of integer compositions of n whose leaders of strictly decreasing runs are strictly increasing.

1, 1, 1, 3, 4, 6, 11, 18, 27, 41, 64, 98, 151, 229, 339, 504, 746, 1097, 1618, 2372, 3451, 5009, 7233, 10394, 14905, 21316, 30396, 43246, 61369, 86830, 122529, 172457, 242092, 339062, 473850, 660829, 919822, 1277935, 1772174, 2453151, 3389762, 4675660, 6438248

Offset: 0

Author

Gus Wiseman, Jul 29 2024

Keywords

• nonn

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 strictly decreasing. The weakly decreasing version is A374764.

Examples

			The a(0) = 1 through a(7) = 18 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)
                (12)  (13)   (14)   (15)   (16)
                (21)  (31)   (23)   (24)   (25)
                      (121)  (32)   (42)   (34)
                             (41)   (51)   (43)
                             (131)  (123)  (52)
                                    (132)  (61)
                                    (141)  (124)
                                    (213)  (142)
                                    (231)  (151)
                                    (321)  (214)
                                           (232)
                                           (241)
                                           (421)
                                           (1213)
                                           (1231)
                                           (1321)
                                           (2131)
		

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

For partitions instead of compositions we have A000009.

The weak version appears to be A188900.

The opposite version is A374689.

Other types of runs (instead of strictly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we have A374634.

- For leaders of anti-runs we have A374679.

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

- For identical leaders we have A374760, ranks A374759.

- For distinct leaders we have A374761, ranks A374767.

- For strictly decreasing leaders we have A374763.

- For weakly increasing leaders we have A374764.

- For weakly decreasing leaders we have A374765.

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.

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

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

Cf. A106356, A188920, A189076, A238343, A261982, A333213, A374518, A374631, A374632, A374687, A374742, A374743.

Programs

Mathematica

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

PARI

seq(n) = Vec(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

Formula

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

Extensions

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