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A332280 Number of integer partitions of n with unimodal run-lengths.

%I A332280 #19 Feb 16 2025 08:33:59
%S A332280 1,1,2,3,5,7,11,15,22,30,41,55,75,97,129,166,215,273,352,439,557,692,
%T A332280 865,1066,1325,1614,1986,2413,2940,3546,4302,5152,6207,7409,8862,
%U A332280 10523,12545,14814,17562,20690,24397,28615,33645,39297,46009,53609,62504,72581,84412
%N A332280 Number of integer partitions of n with unimodal run-lengths.
%C A332280 First differs from A000041 at a(10) = 41, A000041(10) = 42.
%C A332280 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing followed by a weakly decreasing sequence.
%H A332280 Alois P. Heinz, <a href="/A332280/b332280.txt">Table of n, a(n) for n = 0..1000</a>
%H A332280 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332280 The a(10) = 41 partitions (A = 10) are:
%e A332280   (A)     (61111)   (4321)     (3211111)
%e A332280   (91)    (55)      (43111)    (31111111)
%e A332280   (82)    (541)     (4222)     (22222)
%e A332280   (811)   (532)     (42211)    (222211)
%e A332280   (73)    (5311)    (421111)   (2221111)
%e A332280   (721)   (5221)    (4111111)  (22111111)
%e A332280   (7111)  (52111)   (3331)     (211111111)
%e A332280   (64)    (511111)  (3322)     (1111111111)
%e A332280   (631)   (442)     (331111)
%e A332280   (622)   (4411)    (32221)
%e A332280   (6211)  (433)     (322111)
%e A332280 Missing from this list is only (33211).
%p A332280 b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
%p A332280      `if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
%p A332280       j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
%p A332280     end:
%p A332280 a:= n-> b(n$2, 0, true):
%p A332280 seq(a(n), n=0..65);  # _Alois P. Heinz_, Feb 20 2020
%t A332280 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332280 Table[Length[Select[IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,30}]
%t A332280 (* Second program: *)
%t A332280 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 A332280 a[n_] := b[n, n, 0, True];
%t A332280 a /@ Range[0, 65] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A332280 The complement is counted by A332281.
%Y A332280 Heinz numbers of these partitions are the complement of A332282.
%Y A332280 Taking 0-appended first-differences instead of run-lengths gives A332283.
%Y A332280 The normal case is A332577.
%Y A332280 The opposite version is A332638.
%Y A332280 Unimodal compositions are A001523.
%Y A332280 Unimodal normal sequences are A007052.
%Y A332280 Numbers whose unsorted prime signature is unimodal are A332288.
%Y A332280 Cf. A007052, A025065, A072706, A100883, A115981, A227038, A317086, A328509, A329398, A332284, A332285, A332294, A332578, A332579.
%K A332280 nonn
%O A332280 0,3
%A A332280 _Gus Wiseman_, Feb 18 2020