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A375406
Number of integer compositions of n that match the dashed pattern 3-12.
2
0, 0, 0, 0, 0, 0, 1, 4, 14, 41, 110, 278, 673, 1576, 3599, 8055, 17732, 38509, 82683, 175830, 370856, 776723, 1616945, 3348500, 6902905, 14174198, 29004911, 59175625, 120414435, 244468774, 495340191, 1001911626, 2023473267, 4081241473, 8222198324, 16548146045, 33276169507
OFFSET
0,8
COMMENTS
First differs from the non-dashed version A335514 at a(9) = 41, A335514(9) = 42, due to the composition (3,1,3,2).
Also the number of integer compositions of n whose leaders of weakly decreasing runs are not weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is not counted under a(13); also q does not match 3-12. On the other hand, the reverse is (3,1,2,2,1,2,1,1), with maximal weakly decreasing runs ((3,1),(2,2,1),(2,1,1)), with leaders (3,2,2), which are not weakly increasing, so it is counted under a(13); meanwhile it matches 3-12, as required.
FORMULA
a(n>0) = 2^(n-1) - A188900(n).
EXAMPLE
The a(0) = 0 through a(8) = 14 compositions:
. . . . . . (312) (412) (413)
(1312) (512)
(3112) (1412)
(3121) (2312)
(3122)
(3212)
(4112)
(4121)
(11312)
(13112)
(13121)
(31112)
(31121)
(31211)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !LessEqual@@First/@Split[#, GreaterEqual]&]], {n, 0, 15}]
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, z_, ___, x_, y_, ___}/; x<y<z]&]], {n, 0, 15}] (*3-12*)
CROSSREFS
For leaders of identical runs we have A056823.
The complement is counted by A188900.
The non-dashed version is A335514, ranks A335479.
Ranks are positions of non-weakly increasing rows in A374740.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
Counting compositions by number of runs: A238130, A238279, A333755.
A373949 counts compositions by run-compressed sum, opposite A373951.
Sequence in context: A196480 A326008 A196713 * A358587 A237853 A132357
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 22 2024
STATUS
approved