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A227038
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Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.
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38
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1, 1, 1, 3, 4, 7, 13, 19, 30, 44, 71, 98, 147, 205, 294, 412, 575, 783, 1077, 1456, 1957, 2634, 3492, 4627, 6082, 7980, 10374, 13498, 17430, 22451, 28767, 36806, 46803, 59467, 75172, 94839, 119285, 149599, 187031, 233355, 290340, 360327, 446222, 551251, 679524, 835964, 1026210
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OFFSET
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0,4
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LINKS
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Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Composition (combinatorics)
Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Unimodality
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FORMULA
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a(n) ~ c * exp(Pi*sqrt(r*n)) / n, where r = 0.9409240878664458093345791978063..., c = 0.05518035191234679423222212249... - Vaclav Kotesovec, Mar 04 2020
a(n) + A332743(n) = 2^(n - 1). - Gus Wiseman, Mar 05 2020
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EXAMPLE
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There are a(8) = 30 such compositions of 8:
01: [ 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 2 ]
03: [ 1 1 1 1 1 2 1 ]
04: [ 1 1 1 1 2 1 1 ]
05: [ 1 1 1 1 2 2 ]
06: [ 1 1 1 2 1 1 1 ]
07: [ 1 1 1 2 2 1 ]
08: [ 1 1 1 2 3 ]
09: [ 1 1 1 3 2 ]
10: [ 1 1 2 1 1 1 1 ]
11: [ 1 1 2 2 1 1 ]
12: [ 1 1 2 2 2 ]
13: [ 1 1 2 3 1 ]
14: [ 1 1 3 2 1 ]
15: [ 1 2 1 1 1 1 1 ]
16: [ 1 2 2 1 1 1 ]
17: [ 1 2 2 2 1 ]
18: [ 1 2 2 3 ]
19: [ 1 2 3 1 1 ]
20: [ 1 2 3 2 ]
21: [ 1 3 2 1 1 ]
22: [ 1 3 2 2 ]
23: [ 2 1 1 1 1 1 1 ]
24: [ 2 2 1 1 1 1 ]
25: [ 2 2 2 1 1 ]
26: [ 2 2 3 1 ]
27: [ 2 3 1 1 1 ]
28: [ 2 3 2 1 ]
29: [ 3 2 1 1 1 ]
30: [ 3 2 2 1 ]
From Gus Wiseman, Mar 05 2020: (Start)
The a(1) = 1 through a(6) = 13 compositions:
(1) (11) (12) (112) (122) (123)
(21) (121) (221) (132)
(111) (211) (1112) (231)
(1111) (1121) (321)
(1211) (1122)
(2111) (1221)
(11111) (2211)
(11112)
(11121)
(11211)
(12111)
(21111)
(111111)
(End)
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MAPLE
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b:= proc(n, i) option remember;
`if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
add(b(n-i*j, i+1)*(j+1), j=1..n/i))
end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 26 2014
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 1, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], normQ[#]&&unimodQ[#]&]], {n, 0, 10}] (* Gus Wiseman, Mar 05 2020 *)
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CROSSREFS
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Cf. A001523 (unimodal compositions), A001522 (smooth unimodal compositions with first and last part 1), A001524 (unimodal compositions such that each up-step is by at most 1 and first part is 1).
Organizing by length rather than sum gives A007052.
The complement is counted by A332743.
The case of run-lengths of partitions is A332577, with complement A332579.
Compositions covering an initial interval are A107429.
Non-unimodal compositions are A115981.
Cf. A000009, A055932, A072704, A317086, A329766, A332578, A332669, A332670.
Sequence in context: A089374 A029552 A193883 * A189994 A125118 A310008
Adjacent sequences: A227035 A227036 A227037 * A227039 A227040 A227041
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KEYWORD
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nonn
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AUTHOR
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Joerg Arndt, Jun 28 2013
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STATUS
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approved
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