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A227038 Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part. 38
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Unimodality
FORMULA
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
EXAMPLE
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)
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
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.
Sequence in context: A089374 A029552 A193883 * A358914 A189994 A125118
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jun 28 2013
STATUS
approved

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)