A332281 Number of integer partitions of n whose run-lengths are not unimodal.

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

Author

Gus Wiseman, Feb 19 2020

Keywords

• nonn

Comments

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

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Unimodal Sequence

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.

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

Programs

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 *)