login
A332281
Number of integer partitions of n whose run-lengths are not unimodal.
34
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 10, 16, 24, 33, 51, 70, 100, 137, 189, 250, 344, 450, 597, 778, 1019, 1302, 1690, 2142, 2734, 3448, 4360, 5432, 6823, 8453, 10495, 12941, 15968, 19529, 23964, 29166, 35525, 43054, 52173, 62861, 75842, 91013, 109208
OFFSET
0,13
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.
LINKS
Eric Weisstein's World of Mathematics, Unimodal Sequence
EXAMPLE
The a(10) = 1 through a(15) = 10 partitions:
(33211) (332111) (44211) (44311) (55211) (44322)
(3321111) (333211) (433211) (55311)
(442111) (443111) (443211)
(33211111) (3332111) (533211)
(4421111) (552111)
(332111111) (4332111)
(4431111)
(33321111)
(44211111)
(3321111111)
MAPLE
b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
end:
a:= n-> combinat[numbpart](n)-b(n$2, 0, true):
seq(a(n), n=0..65); # Alois P. Heinz, Feb 20 2020
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], !unimodQ[Length/@Split[#]]&]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, j, t && j >= m], {j, 1, Min[If[t, Infinity, m], n/i]}] + b[n, i - 1, m, t]]];
a[n_] := PartitionsP[n] - b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
CROSSREFS
The complement is counted by A332280.
The Heinz numbers of these partitions are A332282.
The opposite version is A332639.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Sequence in context: A279715 A132212 A327047 * A241903 A261204 A374145
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 19 2020
STATUS
approved