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%I #46 Sep 16 2023 19:06:03
%S 1,2,2,2,6,4,10,10,4,32,26,8,68,64,34,8,220,186,82,16,528,488,276,98,
%T 16,1724,1484,744,226,32,4460,4086,2382,980,258,32,14664,12752,6822,
%U 2498,578,64,39908,36384,21616,9576,3088,642,64,131944,115508,64264,26040,7552,1410,128
%N Irregular triangle read by rows: T(n,k) is the number of degree-n permutations without overlaps which furnish k new permutations without overlaps upon the addition of an (n+1)st element, 2 <= k <= 1 + floor(n/2).
%C See Sade for precise definition.
%C From _Roger Ford_, Dec 07 2018: (Start)
%C T(n,k) is the number of semi-meanders with n top arches, k top arch groupings and a rainbow of bottom arches.
%C Example: /\ /\
%C n=4 k=3 //\\ /\ /\, /\ /\ //\\ T(4,3) = 2
%C .
%C /\ /\
%C //\\ //\\
%C n=4 k=2 ///\\\ /\, /\ ///\\\ T(4,2) = 2. (End)
%C Stéphane Legendre's solutions for folding a strip of stamps with leaf 1 on top have the same numeric sequences and total solutions as Albert Sade's permutations without overlaps. Stéphane Legendre's "Illustration of initial terms" link in A000682 models the values for Albert Sade's array. - _Roger Ford_, Dec 24 2018
%D A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949.
%H Andrew Howroyd, <a href="/A259689/b259689.txt">Table of n, a(n) for n = 2..170</a>
%H Albert Sade, <a href="/A000108/a000108_17.pdf">Sur les Chevauchements des Permutations</a>, published by the author, Marseille, 1949. [Annotated scanned copy]
%F Sum_{k>=2} k*T(n,k) = A000682(n + 1). - _Andrew Howroyd_, Dec 07 2018
%F T(n, floor(n/2)) = 2^floor((n-1)/2)*(n-4)+2. - _Roger Ford_, Dec 04 2018
%F For n>2, T(n, floor((n+2)/2)) = 2^(floor((n-1)/2)). - _Roger Ford_, Aug 18 2023
%e Triangle begins, n >= 2, 2 <= k <= 1 + floor(n/2):
%e 1;
%e 2;
%e 2, 2;
%e 6, 4;
%e 10, 10, 4;
%e 32, 26, 8;
%e 68, 64, 34, 8;
%e 220, 186, 82, 16;
%e 528, 488, 276, 98, 16;
%e 1724, 1484, 744, 226, 32;
%e 4460, 4086, 2382, 980, 258, 32;
%e ...
%Y Row sums give A000682.
%Y Column k=2 is A260785.
%K nonn,tabf
%O 2,2
%A _N. J. A. Sloane_, Jul 04 2015
%E Terms a(22) and beyond from _Andrew Howroyd_, Dec 05 2018
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