%I #27 Jan 28 2019 13:57:52
%S 1,2,2,6,10,32,68,220,528,1724,4460,14664,39908,131944,372448,1237016,
%T 3589384,11967140,35479312,118675768,357957984,1200724776,3673173656,
%U 12351611656,38232022416
%N Leading diagonal of triangle in A259689: a(n)= number of permutations without overlaps that generate exactly 2 permutations without overlaps in a(n+1).
%C From _Roger Ford_, Oct 12 2015: (Start)
%C a(n)= Number of semi-meander solutions for n with 2 returns to the x axis (or number of 2 distinct arch groups).
%C Example: n=5 -= return to x axis
%C /\ /\ /\
%C //\\ / \ /\ //\\
%C ///\\\ / /\\ /\ //\\ ///\\\
%C /\-////\\\\- /\-//\//\\\- //\\-///\\\- ////\\\\-/\-
%C /\
%C / \ /\
%C //\ \ //\\ /\
%C ///\\/\\-/\- ///\\\-//\\- a(5)=6.
%C a(n)= Number of hills (arches with a peak at 1 and no covering arches) for semi-meander solutions with n-1 arches.
%C Example: n=5 semi-meander solutions with 4 arches (/\)= hill
%C /\ /\
%C /\ /\ //\\ //\\
%C (/\)(/\)//\\ //\\(/\)(/\) ///\\\(/\) (/\)///\\\ a(5)=6.
%C (End)
%C From _Roger Ford_, Jan 27 2018: (Start)
%C a(n)= Number of solutions for folding a strip of n stamps with stamp 1 on top and each solution ordering having the absolute value of the difference of the stamp number before and after stamp n equal to 1. (If stamp n is the last stamp in the solution ordering then add a(1) to the end of the ordering.)
%C Example: n=5
%C 12354 |3-4| = 1, 14325(1) |2-1| = 1, 12453 |4-3| = 1,
%C 14532 |4-3| = 1, 15234 |1-2| = 1, 13542 |3-4| = 1, a(5)=6.
%C (End)
%H Albert Sade, <a href="/A000108/a000108_17.pdf">Sur les Chevauchements des Permutations</a>, published by the author, Marseille, 1949. [Annotated scanned copy]
%Y Column k=2 of A259689.
%K nonn,more
%O 2,2
%A _N. J. A. Sloane_, Aug 04 2015
%E Corrected and extended by _Roger Ford_, Oct 12 2015
%E a(14)-(26) from _Andrew Howroyd_, Dec 05 2018
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