A332280 - OEIS

A332280

Number of integer partitions of n with unimodal run-lengths.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 97, 129, 166, 215, 273, 352, 439, 557, 692, 865, 1066, 1325, 1614, 1986, 2413, 2940, 3546, 4302, 5152, 6207, 7409, 8862, 10523, 12545, 14814, 17562, 20690, 24397, 28615, 33645, 39297, 46009, 53609, 62504, 72581, 84412

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OFFSET

0,3

COMMENTS

First differs from A000041 at a(10) = 41, A000041(10) = 42.

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Unimodal Sequence

EXAMPLE

The a(10) = 41 partitions (A = 10) are:

(A) (61111) (4321) (3211111)

(91) (55) (43111) (31111111)

(82) (541) (4222) (22222)

(811) (532) (42211) (222211)

(73) (5311) (421111) (2221111)

(721) (5221) (4111111) (22111111)

(7111) (52111) (3331) (211111111)

(64) (511111) (3322) (1111111111)

(631) (442) (331111)

(622) (4411) (32221)

(6211) (433) (322111)

Missing from this list is only (33211).

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-> 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_] := 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 A332281.

Heinz numbers of these partitions are the complement of A332282.

Taking 0-appended first-differences instead of run-lengths gives A332283.

The normal case is A332577.

The opposite version is A332638.

Unimodal compositions are A001523.

Unimodal normal sequences are A007052.

Numbers whose unsorted prime signature is unimodal are A332288.