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%I #11 May 19 2020 19:11:24
%S 1,1,2,4,7,12,19,30,46,69,102,149,214,304,428,596,823,1127,1532,2068,
%T 2774
%N Number of compositions of n whose non-adjacent parts are weakly decreasing.
%e The a(1) = 1 through a(6) = 19 compositions:
%e (1) (2) (3) (4) (5) (6)
%e (11) (12) (13) (14) (15)
%e (21) (22) (23) (24)
%e (111) (31) (32) (33)
%e (121) (41) (42)
%e (211) (131) (51)
%e (1111) (212) (141)
%e (221) (222)
%e (311) (231)
%e (1211) (312)
%e (2111) (321)
%e (11111) (411)
%e (1212)
%e (1311)
%e (2121)
%e (2211)
%e (3111)
%e (12111)
%e (21111)
%e (111111)
%e For example, (2,3,1,2) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), all of which are weakly decreasing.
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,__,y_,___}/;y>x]&]],{n,0,15}]
%Y Unimodal compositions are A001523.
%Y The case of normal sequences appears to be A028859.
%Y A version for ordered set partitions is A332872.
%Y The case of strict compositions is A333150.
%Y The version for strictly decreasing parts is A333193.
%Y Standard composition numbers (A066099) of these compositions are A334966.
%Y Cf. A056242, A059204, A072706, A107429, A115981, A329398, A332578, A332669, A332673, A332724, A332834.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, May 16 2020
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