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A332871
Number of compositions of n whose run-lengths are not weakly increasing.
2
0, 0, 0, 0, 1, 4, 8, 24, 55, 128, 282, 625, 1336, 2855, 6000, 12551, 26022, 53744, 110361, 225914, 460756, 937413, 1902370, 3853445, 7791647, 15732468, 31725191, 63907437, 128613224, 258626480, 519700800, 1043690354, 2094882574, 4202903667, 8428794336, 16897836060
OFFSET
0,6
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
Also compositions whose run-lengths are not weakly decreasing.
LINKS
FORMULA
a(n) = 2^(n - 1) - A332836(n).
EXAMPLE
The a(4) = 1 through a(6) = 8 compositions:
(112) (113) (114)
(221) (1113)
(1112) (1131)
(1121) (1221)
(2112)
(11112)
(11121)
(11211)
For example, the composition (2,1,1,2) has run-lengths (1,2,1), which are not weakly increasing, so (2,1,1,2) is counted under a(6).
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !LessEqual@@Length/@Split[#]&]], {n, 0, 10}]
CROSSREFS
The version for the compositions themselves (not run-lengths) is A056823.
The version for unsorted prime signature is A112769, with dual A071365.
The case without weakly decreasing run-lengths either is A332833.
The complement is counted by A332836.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are not unimodal are A332727.
Sequence in context: A319721 A115641 A153334 * A116719 A159612 A099176
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 29 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020
STATUS
approved