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A333193 Number of compositions of n whose non-adjacent parts are strictly decreasing. 5
1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 71, 93, 122, 158, 204, 260, 332, 419, 528, 661, 825, 1023, 1267, 1560, 1916, 2344, 2860, 3476, 4217, 5097, 6147, 7393, 8872, 10618, 12685, 15115, 17977, 21336, 25276, 29882, 35271, 41551, 48872, 57385, 67277, 78745, 92040 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
EXAMPLE
The a(1) = 1 through a(7) = 15 compositions:
(1) (2) (3) (4) (5) (6) (7)
(11) (12) (13) (14) (15) (16)
(21) (22) (23) (24) (25)
(31) (32) (33) (34)
(211) (41) (42) (43)
(221) (51) (52)
(311) (231) (61)
(312) (241)
(321) (322)
(411) (331)
(2211) (412)
(421)
(511)
(2311)
(3211)
For example, (2,3,1,2) is not such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), not all of which are strictly decreasing, while (2,4,1,1) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,1), (4,1), all of which are strictly decreasing.
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, x_, __, y_, ___}/; y>=x]&]], {n, 0, 15}]
PROG
(PARI) \\ p is all, q is those ending in an unreversed singleton.
seq(n)={my(q=O(x*x^n), p=1+q); for(k=1, n, [p, q] = [p*(1 + x^k + x^(2*k)) + q*x^k, q + p*x^k] ); Vec(p)} \\ Andrew Howroyd, Apr 17 2021
CROSSREFS
A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The case of normal sequences appears to be A001045.
Unimodal compositions are A001523, with strict case A072706.
Strict compositions are A032020.
Partitions with strictly increasing run-lengths are A100471.
Partitions with strictly decreasing run-lengths are A100881.
Compositions with weakly decreasing non-adjacent parts are A333148.
Compositions with strictly increasing run-lengths are A333192.
Sequence in context: A035977 A288256 A101049 * A035986 A035996 A261775
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 18 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Apr 17 2021
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)