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A333193
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Number of compositions of n whose non-adjacent parts are strictly decreasing.
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5
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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
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OFFSET
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0,3
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, x_, __, y_, ___}/; y>=x]&]], {n, 0, 15}]
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PROG
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(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
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CROSSREFS
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A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The case of normal sequences appears to be A001045.
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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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