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A000682 Semimeanders: number of ways a semi-infinite directed curve can cross a straight line n times.
(Formerly M1205 N0464)
11
1, 1, 2, 4, 10, 24, 66, 174, 504, 1406, 4210, 12198, 37378, 111278, 346846, 1053874, 3328188, 10274466, 32786630, 102511418, 329903058, 1042277722, 3377919260, 10765024432, 35095839848, 112670468128, 369192702554, 1192724674590, 3925446804750 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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 1-2n 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)

REFERENCES

A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Touchard, Contributions a` l'e'tude du proble`me des timbres postes, Canad. J. Math., 2 (1950), 385-398.

LINKS

I. Jensen, Table of n, a(n) for n = 1..45

P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics.

P. Di Francesco, O. Golinelli and E. Guitter, Meanders: a direct enumeration approach, Nucl. Phys. B 482 [ FS ] (1996) 497-535.

P. Di Francesco, Matrix model combinatorics: applications to folding and coloring

I. Jensen, Home page

I. Jensen, Terms a(1)..a(45)

I. Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).

I. Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).

J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152.

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 map-folding problem, Math. Comp. 22 (1968), 193-199.

J. Sawada and R. Li, Stamp foldings, semi-meanders, 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

EXAMPLE

a(3) = 4: the four solutions with three crossings are the two solutions shown in A086441 together with their reflections about a North-South axis.

CROSSREFS

A000560(n) = a(n)/2 (for n >= 2) gives number of nonisomorphic solutions (see also A086441). Cf. A001011, A001997.

Sequence in context: A084078 A137842 A049146 * A001997 A239605 A000084

Adjacent sequences:  A000679 A000680 A000681 * A000683 A000684 A000685

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Sade gives the first 11 terms. Computed to n = 45 by Iwan Jensen.

STATUS

approved

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Last modified October 21 11:15 EDT 2014. Contains 248377 sequences.