A332280 - OEIS

A332280

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

%I #19 Feb 16 2025 08:33:59

%S 1,1,2,3,5,7,11,15,22,30,41,55,75,97,129,166,215,273,352,439,557,692,

%T 865,1066,1325,1614,1986,2413,2940,3546,4302,5152,6207,7409,8862,

%U 10523,12545,14814,17562,20690,24397,28615,33645,39297,46009,53609,62504,72581,84412

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

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

%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="/A332280/b332280.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%e (A) (61111) (4321) (3211111)

%e (91) (55) (43111) (31111111)

%e (82) (541) (4222) (22222)

%e (811) (532) (42211) (222211)

%e (73) (5311) (421111) (2221111)

%e (721) (5221) (4111111) (22111111)

%e (7111) (52111) (3331) (211111111)

%e (64) (511111) (3322) (1111111111)

%e (631) (442) (331111)

%e (622) (4411) (32221)

%e (6211) (433) (322111)

%e Missing from this list is only (33211).

%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-> 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_] := 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 A332281.

%Y Heinz numbers of these partitions are the complement of A332282.

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

%Y The normal case is A332577.

%Y The opposite version is A332638.

%Y Unimodal compositions are A001523.

%Y Unimodal normal sequences are A007052.

%Y Numbers whose unsorted prime signature is unimodal are A332288.

%Y Cf. A007052, A025065, A072706, A100883, A115981, A227038, A317086, A328509, A329398, A332284, A332285, A332294, A332578, A332579.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 18 2020