A230439 Number of contractible "tight" meanders of width n.

1, 2, 6, 14, 34, 68, 150, 296, 586, 1140, 2182, 4130, 7678, 14368, 26068, 48248, 86572, 158146, 281410, 509442, 901014, 1618544, 2852464, 5089580, 8948694, 15884762, 27882762, 49291952, 86435358, 152316976, 266907560, 469232204, 821844316

Offset: 1

Author

Mamuka Jibladze, Nov 04 2013

Keywords

• nonn

Comments

A tight meander of width n is a special kind of meander defined as follows.

For any pair (S={s_1,...,s_k},T={t_1,...,t_l}) of subsets of {1,...,n-1} (k or l might be 0), the tight meander M(S,T) defined by (S,T) is the following subset of R^2:

assuming S and T ordered so that 0=s_0

semicircles in the upper half-plane with endpoints (s_{i-1}+j,0) and (s_i+1-j,0), for i=1,...,k+1, and j positive integer with s_{i-1}+j

and semicircles in the lower half-plane with endpoints (t_{i-1}+j,0) and (t_i+1-j,0), for i=1,...,l+1, and j positive integer with t_{i-1}+j

The tight meander M(S,T) is called contractible if it is a contractible subspace of R^2, i.e., is either a single point or homeomorphic to an interval.

Then, a(n) is the number of pairs (S,T) as above such that the tight meander M(S,T) is contractible.

From Roger Ford, Jul 05 2023: (Start)

The following is a definition for closed meanders that yield the same sequence as tight meanders. T(n,k) = the number of closed meanders with n top arches and with k exterior arches and k arches of length 1.

e = exterior arch (arch with no covering arch), 1 = arch with length 1, e1 = arch that is exterior with a length of 1:

e exterior length 1

__________ arches arches

/ ____ \

e1 / / \ \ top = 2 top = 2

/\ / / /\1 \ \

/ \ / / / \ \ \

\ \ / / \ \ / / bottom = 2 bottom = 2

\ \/1 / \ \/1 / total = 4 total = 4

\____/ \____/

e e Example T(4,4).

(End)

Examples

			For n=3 the a(3)=6 contractible tight meanders of width 3 correspond to the following pairs of subsets of {1,2}: ({},{1}), ({},{2}), ({1},{}), ({2},{}), ({1},{2}), ({2},{1}).
		

Links

• Mamuka Jibladze, Table of n, a(n) for n = 1..100 (first 64 terms by Martin Plechsmid)
• Vincent Coll, Colton Magnant, and Hua Wang, The Signature of a Meander, arXiv:1206.2705 [math.QA], 2012.
• Vladimir Dergachev and Alexandre Kirillov, Index of Lie algebras of Seaweed Type, J. Lie Theory, 10 (2000), 331-343
• Mathoverflow, "Special" meanders
• Dmitri I. Panyushev, Inductive Formulas for the Index of Seaweed Lie Algebras, Moscow Math. J., 1 (2001), 221-241.

Crossrefs

For various kinds of meandric numbers see A005315, A005316, A060066, A060089, A060206.

Programs

Maple

# program based on the C code by Martin Plechsmid:
proc()
local n,a,b,d,r;
option remember;
  if args[1]=1 then
   1
  elif nargs=1 then
   2*`+`(''procname(args,[i],[j])'$'j'=1..i-1'$'i'=2..args)
  else
   n:=args[1]; a:=args[2]; b:=args[3];
   if b=[] then
    `+`('procname(n,a,[k])'$'k'=1..n)
   elif a[1]=b[1] then
    0
   elif a[1]0 then
     procname(n-b[1],[d-r,op(subsop(1=r,a))],subsop(1=NULL,b))
    else
     procname(n-b[1],subsop(1=d,a),subsop(1=NULL,b))
    fi
   fi
  fi
end;

Mathematica

(* program based on the C code by Martin Plechsmid: *)
f[n_,a_,b_]:=Which[
n==1, 1,
b=={}, f[n,a,b]=Sum[f[n,a,{i}],{i,n}],
a=={} || First[a]
				

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