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A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

%I A332835 #19 Feb 16 2025 08:33:59
%S A332835 1,1,2,4,8,16,29,56,101,181,327,583,1023,1820,3207,5631,9905,17394,
%T A332835 30489,53481,93725,164169,287606,503672,881834,1544018,2703161,
%U A332835 4731860,8283291,14499392,25379278,44422866,77754798,136093756,238204369,416923752,729728031
%N A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.
%C A332835 A composition of n is a finite sequence of positive integers summing to n.
%H A332835 Andrew Howroyd, <a href="/A332835/b332835.txt">Table of n, a(n) for n = 0..1000</a>
%H A332835 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332835 a(n) = 2 * A332836(n) - A329738(n).
%e A332835 The a(6) = 29 compositions:
%e A332835   (6)    (141)  (213)   (1113)  (21111)
%e A332835   (51)   (114)  (132)   (222)   (12111)
%e A332835   (15)   (33)   (123)   (2211)  (11121)
%e A332835   (42)   (321)  (3111)  (2121)  (11112)
%e A332835   (24)   (312)  (1311)  (1212)  (111111)
%e A332835   (411)  (231)  (1131)  (1122)
%e A332835 Missing are: (2112), (1221), (11211).
%t A332835 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]
%Y A332835 The version for the compositions themselves (not run-lengths) is A329398.
%Y A332835 Compositions with equal run-lengths are A329738.
%Y A332835 The case of partitions is A332745.
%Y A332835 The version for unsorted prime signature is the complement of A332831.
%Y A332835 The complement is counted by A332833.
%Y A332835 Unimodal compositions are A001523.
%Y A332835 Partitions with weakly decreasing run-lengths are A100882.
%Y A332835 Partitions with weakly increasing run-lengths are A100883.
%Y A332835 Compositions that are not unimodal are A115981.
%Y A332835 Compositions whose negation is unimodal are A332578.
%Y A332835 Compositions whose run-lengths are unimodal are A332726.
%Y A332835 Neither weakly increasing nor weakly decreasing compositions are A332834.
%Y A332835 Compositions with weakly increasing run-lengths are A332836.
%Y A332835 Compositions that are neither unimodal nor is their negation are A332870.
%Y A332835 Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.
%K A332835 nonn
%O A332835 0,3
%A A332835 _Gus Wiseman_, Feb 29 2020
%E A332835 Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2020