cp's OEIS Frontend

A333149 Number of strict compositions of n that are neither increasing nor decreasing.

%I A333149 #6 Feb 16 2025 08:33:59
%S A333149 0,0,0,0,0,0,4,4,8,12,38,42,72,98,150,298,372,542,760,1070,1428,2600,
%T A333149 3120,4550,6050,8478,10976,15220,23872,29950,41276,55062,74096,97148,
%U A333149 129786,167256,256070,314454,429338,556364,749266,955746,1275016,1618054
%N A333149 Number of strict compositions of n that are neither increasing nor decreasing.
%C A333149 A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.
%H A333149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A333149 a(n) = A032020(n) - 2*A000009(n) + 1.
%e A333149 The a(6) = 4 through a(9) = 12 compositions:
%e A333149   (1,3,2)  (1,4,2)  (1,4,3)  (1,5,3)
%e A333149   (2,1,3)  (2,1,4)  (1,5,2)  (1,6,2)
%e A333149   (2,3,1)  (2,4,1)  (2,1,5)  (2,1,6)
%e A333149   (3,1,2)  (4,1,2)  (2,5,1)  (2,4,3)
%e A333149                     (3,1,4)  (2,6,1)
%e A333149                     (3,4,1)  (3,1,5)
%e A333149                     (4,1,3)  (3,2,4)
%e A333149                     (5,1,2)  (3,4,2)
%e A333149                              (3,5,1)
%e A333149                              (4,2,3)
%e A333149                              (5,1,3)
%e A333149                              (6,1,2)
%t A333149 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!Greater@@#&&!Less@@#&]],{n,0,10}]
%Y A333149 The non-strict case is A332834.
%Y A333149 The complement is counted by A333147.
%Y A333149 Strict partitions are A000009.
%Y A333149 Strict compositions are A032020.
%Y A333149 Non-unimodal strict compositions are A072707.
%Y A333149 Strict partitions with increasing or decreasing run-lengths are A333190.
%Y A333149 Strict compositions with increasing or decreasing run-lengths are A333191.
%Y A333149 Unimodal compositions are A001523, with strict case A072706.
%Y A333149 Cf. A059204, A115981, A227038, A329398, A332745, A332746, A332831, A332833, A332835, A332874, A333150, A333192.
%K A333149 nonn
%O A333149 0,7
%A A333149 _Gus Wiseman_, May 16 2020