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A259689 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).

%I A259689 #46 Sep 16 2023 19:06:03
%S A259689 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 A259689 16,1724,1484,744,226,32,4460,4086,2382,980,258,32,14664,12752,6822,
%U A259689 2498,578,64,39908,36384,21616,9576,3088,642,64,131944,115508,64264,26040,7552,1410,128
%N A259689 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 A259689 See Sade for precise definition.
%C A259689 From _Roger Ford_, Dec 07 2018: (Start)
%C A259689 T(n,k) is the number of semi-meanders with n top arches, k top arch groupings and a rainbow of bottom arches.
%C A259689 Example:     /\                      /\
%C A259689 n=4 k=3     //\\ /\ /\,       /\ /\ //\\    T(4,3) = 2
%C A259689 .
%C A259689                /\                  /\
%C A259689               //\\                //\\
%C A259689 n=4 k=2      ///\\\ /\,       /\ ///\\\     T(4,2) = 2. (End)
%C A259689 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 A259689 A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949.
%H A259689 Andrew Howroyd, <a href="/A259689/b259689.txt">Table of n, a(n) for n = 2..170</a>
%H A259689 Albert Sade, <a href="/A000108/a000108_17.pdf">Sur les Chevauchements des Permutations</a>, published by the author, Marseille, 1949. [Annotated scanned copy]
%F A259689 Sum_{k>=2} k*T(n,k) = A000682(n + 1). - _Andrew Howroyd_, Dec 07 2018
%F A259689 T(n, floor(n/2)) = 2^floor((n-1)/2)*(n-4)+2. - _Roger Ford_, Dec 04 2018
%F A259689 For n>2, T(n, floor((n+2)/2)) = 2^(floor((n-1)/2)). - _Roger Ford_, Aug 18 2023
%e A259689 Triangle begins, n >= 2, 2 <= k <= 1 + floor(n/2):
%e A259689      1;
%e A259689      2;
%e A259689      2,    2;
%e A259689      6,    4;
%e A259689     10,   10,    4;
%e A259689     32,   26,    8;
%e A259689     68,   64,   34,   8;
%e A259689    220,  186,   82,  16;
%e A259689    528,  488,  276,  98,  16;
%e A259689   1724, 1484,  744, 226,  32;
%e A259689   4460, 4086, 2382, 980, 258, 32;
%e A259689   ...
%Y A259689 Row sums give A000682.
%Y A259689 Column k=2 is A260785.
%K A259689 nonn,tabf
%O A259689 2,2
%A A259689 _N. J. A. Sloane_, Jul 04 2015
%E A259689 Terms a(22) and beyond from _Andrew Howroyd_, Dec 05 2018