
COMMENTS

Number of permutations of n+1 letters without overlaps [Sade, 1949].  N. J. A. Sloane, Jul 05 2015
Number of ways to fold a strip of n+1 labeled stamps with leaf 1 on top. [Clarified by Stéphane Legendre, Apr 09 2013]
From Roger Ford, Jul 04 2014: (Start)
The number of semimeander solutions for n (a(n)) is equal to the number of n top arch solutions in the intersection of A001263 (with no intersecting top arches) and A244312 (arches forming a complete loop).
The top and bottom arches for semimeanders pass through vertices 12n on a straight line with the arches below the line forming a rainbow pattern.
The number of total arches going from an odd vertex to a higher even vertex must be exactly 2 greater than the number of arches going from an even vertex to a higher odd vertex to form a single complete loop with no intersections.
The arch solutions in the intersection of A001263 (T(n,k)) and A244312 (F(n,k)) occur when the number of top arches going from a odd vertex to a higher even vertex (k) meets the condition that k = ceiling((n+1)/2).
Example: semimeanders a(5)=10.
(A244312) F(5,3)=16 { 10 commom solutions: [12,34,5 10,67,89] [16,23,45,78,9 10] [12,36,45,7 10,89] [14,23,58,67,9 10] [12,3 10,49,58,67] [18,27,36,45,9 10] [12,3 10,45,69,78] [18,25,34,67,9 10] [14,23,5 10,69,78] [16,25,34,7 10,89] } + [18,27,34,5 10,69] [16,25,3 10,49,78] [18,25,36,49,7 10] [14,27,3 10,58,69] [14,27,36,5 10,89] [16,23,49,58,7 10]
(A001263) T(5,3)=20 { 10 common solutions } + [12,38,45,67,9 10] [1 10,29,38,47,56] [1 10,25,34,69,78] [14,23,56,7 10,89] [12,3 10,47,56,89] [18,23,47,56,9 10] [1 10,29,36,45,78] [1 10,29,34,58,67] [1 10,27,34,56,89] [1 10,23,49,56,78].
(End)
From Roger Ford, May 10 2015: (Start)
Conjecture: The number of semi meander solutions for n arches (M(n)) is equal to the sum of all the returns to the xaxis for semi meander solutions with n1 arches (M(n1)) written in binary form.
Example: 1= arches up from xaxis, 0= arches down to x axis, _= return to xaxis
M(4)=4 /\ /\
//\\/\/\ = 1100_10_10_ 3 returns /\/\//\\ = 10_10_1100_ 3 returns
/\ /\
//\\ //\\
///\\\/\ = 111000_10_ 2 returns /\///\\\ = 10_111000_ 2 returns
sum of returns=10, M(5)=10. (End)


LINKS

I. Jensen, Table of n, a(n) for n = 1..45
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P. Di Francesco, Matrix model combinatorics: applications to folding and coloring
I. Jensen, Home page
I. Jensen, Terms a(1)..a(45)
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J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135152.
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135152. [Annotated, corrected, scanned copy]
M. La Croix, Approaches to the Enumerative Theory of Meanders
Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025, 2013.
Stéphane Legendre, Illustration of initial terms
W. F. Lunnon, A mapfolding problem, Math. Comp. 22 (1968), 193199.
A. Panayotopoulos, P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 110.
Albert Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949. [Annotated scanned copy]
J. Sawada and R. Li, Stamp foldings, semimeanders, and open meanders: fast generation algorithms, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).
Index entries for sequences obtained by enumerating foldings
