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

0,4

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.

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

Example

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)

Mathematica

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

Prog

(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

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.

Sequence in context: A069825 A093040 A022935 * A192813 A048229 A002090

Adjacent sequences: A374759 A374760 A374761 * A374763 A374764 A374765

Keyword

nonn

Author

Gus Wiseman, Jul 29 2024

Extensions

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

Status

approved