A332281 - OEIS

A332281

Number of integer partitions of n whose run-lengths are not unimodal.

%I #15 May 10 2021 04:01:29

%S 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,

%T 450,597,778,1019,1302,1690,2142,2734,3448,4360,5432,6823,8453,10495,

%U 12941,15968,19529,23964,29166,35525,43054,52173,62861,75842,91013,109208

%N Number of integer partitions of n whose run-lengths are not unimodal.

%C A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.

%H Alois P. Heinz, <a href="/A332281/b332281.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%e The a(10) = 1 through a(15) = 10 partitions:

%e (33211) (332111) (44211) (44311) (55211) (44322)

%e (3321111) (333211) (433211) (55311)

%e (442111) (443111) (443211)

%e (33211111) (3332111) (533211)

%e (4421111) (552111)

%e (332111111) (4332111)

%e (4431111)

%e (33321111)

%e (44211111)

%e (3321111111)

%p b:= proc(n, i, m, t) option remember; `if`(n=0, 1,

%p `if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),

%p j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))

%p end:

%p a:= n-> combinat[numbpart](n)-b(n$2, 0, true):

%p seq(a(n), n=0..65); # _Alois P. Heinz_, Feb 20 2020

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]

%t Table[Length[Select[IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,30}]

%t (* Second program: *)

%t 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]]];

%t a[n_] := PartitionsP[n] - b[n, n, 0, True];

%t a /@ Range[0, 65] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)

%Y The complement is counted by A332280.

%Y The Heinz numbers of these partitions are A332282.

%Y The opposite version is A332639.

%Y Unimodal compositions are A001523.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Non-unimodal normal sequences are A328509.

%Y Cf. A007052, A025065, A072706, A100883, A332283, A332284, A332286, A332287, A332579, A332638, A332640, A332641, A332642.

%K nonn

%O 0,13

%A _Gus Wiseman_, Feb 19 2020