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 A332280 Number of integer partitions of n with unimodal run-lengths. 34
 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 (list; graph; refs; listen; history; text; internal format)
 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[]<=q[], 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. Cf. A007052, A025065, A072706, A100883, A115981, A227038, A317086, A328509, A329398, A332284, A332285, A332294, A332578, A332579. Sequence in context: A023030 A246580 A212187 * A035998 A137792 A039905 Adjacent sequences:  A332277 A332278 A332279 * A332281 A332282 A332283 KEYWORD nonn AUTHOR Gus Wiseman, Feb 18 2020 STATUS approved

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Last modified August 5 21:45 EDT 2021. Contains 346488 sequences. (Running on oeis4.)