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%I #14 Jan 12 2024 00:49:37
%S 1,0,1,0,1,1,0,1,2,1,0,1,3,2,1,0,1,4,5,2,1,0,1,5,7,5,2,1,0,1,6,11,10,
%T 5,2,1,0,1,7,15,16,10,5,2,1,0,1,8,20,24,20,10,5,2,1,0,1,9,25,36,31,20,
%U 10,5,2,1,0,1,10,32,50,50,36,20,10,5,2,1
%N Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.
%C A composition of n is a finite sequence of positive integers summing to n.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H Andrew Howroyd, <a href="/A332670/b332670.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F G.f.: A(x,y) = 1 + Sum_{j>0} y*x^j/((1 - y*x^j)*Product_{k>j} (1 - y*x^k)^2). - _Andrew Howroyd_, Jan 11 2024
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 2 1
%e 0 1 3 2 1
%e 0 1 4 5 2 1
%e 0 1 5 7 5 2 1
%e 0 1 6 11 10 5 2 1
%e 0 1 7 15 16 10 5 2 1
%e 0 1 8 20 24 20 10 5 2 1
%e 0 1 9 25 36 31 20 10 5 2 1
%e 0 1 10 32 50 50 36 20 10 5 2 1
%e 0 1 11 38 67 73 59 36 20 10 5 2 1
%e Column n = 7 counts the following compositions:
%e (7) (16) (115) (1114) (11113) (111112) (1111111)
%e (25) (124) (1123) (11122) (211111)
%e (34) (133) (1222) (21112)
%e (43) (214) (2113) (22111)
%e (52) (223) (2122) (31111)
%e (61) (313) (2212)
%e (322) (2221)
%e (331) (3112)
%e (412) (3211)
%e (421) (4111)
%e (511)
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],unimodQ[-#]&]],{n,0,10},{k,0,n}]
%o (PARI) T(n)={[Vecrev(p) | p<-Vec(1 + sum(j=1, n, y*x^j/((1-y*x^j) * prod(k=j+1, n-j, 1 - y*x^k + O(x*x^(n-j)))^2)))]}
%o { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 11 2024
%Y The case of partitions is A072233.
%Y Dominated by A072704 (the non-negated version).
%Y The strict case is A072705.
%Y The case of constant compositions is A113704.
%Y Row sums are A332578.
%Y Unimodal compositions are A001523.
%Y Unimodal normal sequences appear to be A007052.
%Y Non-unimodal compositions are A115981.
%Y Non-unimodal normal sequences are A328509.
%Y Numbers whose negated unsorted prime signature is not unimodal are A332282.
%Y Partitions whose negated run-lengths are unimodal are A332638.
%Y Compositions whose negation is not unimodal are A332669.
%Y Partitions whose negated 0-appended first differences are unimodal: A332728.
%Y Cf. A011782, A107429, A227038, A332280, A332283, A332639, A332642, A332741, A332742, A332744, A332832, A332870.
%K nonn,tabl
%O 0,9
%A _Gus Wiseman_, Feb 29 2020
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