

A374762


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


11



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
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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



FORMULA

G.f.: Product_{k>=1} (1 + x^k*Product_{j=1..k1} (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(nk, k1), 1 + x^j, 1 + O(x^(nk+1))))) \\ Andrew Howroyd, Jul 31 2024


CROSSREFS

For partitions instead of compositions we have A000009.
The weak version appears to be A188900.
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 antiruns we have A374679.
Other types of runleaders (instead of strictly increasing):
 For strictly decreasing leaders we have A374763.
 For weakly increasing leaders we have A374764.
 For weakly decreasing leaders we have A374765.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A106356, A188920, A189076, A238343, A261982, A333213, A374518, A374631, A374632, A374687, A374742, A374743.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



