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A230439 Number of contractible "tight" meanders of width n. 1
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 (list; graph; refs; listen; history; text; internal format)



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<s_1<...<s_k<s_{k+1}=n and 0=t_0<t_1<...<t_l<t_{l+1}=n, let M(S,T) be the union of the set of points {(1,0),...,(n,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<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<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.


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 preprint 1206.2705

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


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}).


# The Maple program based on the C code by Martin Plechsmid


local n, a, b, d, r;

option remember;

  if args[1]=1 then


  elif nargs=1 then

   2*`+`(''procname(args, [i], [j])'$'j'=1..i-1'$'i'=2..args)


   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


   elif a[1]<b[1] then

    procname(n, b, a)



    r:=irem(b[1], d);

    if r>0 then

     procname(n-b[1], [d-r, op(subsop(1=r, a))], subsop(1=NULL, b))


     procname(n-b[1], subsop(1=d, a), subsop(1=NULL, b))






(* Mathematica program based on the C code by Martin Plechsmid *)

f[n_, a___, b___] := f[n, a, b] =


   n == 1, 1,

   a == Null, 2 Sum[f[n, {i}, {j}], {i, 2, n}, {j, i - 1}],

   b == {}, Sum[f[n, a, {i}], {i, n}],

   First[a] == First[b], 0,

   First[a] < First[b], f[n, b, a],


   Block[{d = First[a] - First[b], r, s},

    r = Mod[First[b], d];

    s = If[r == 0, {d}, {d - r, r}];

    f[n - First[b], Join[s, Rest[a]], Rest[b]]




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

Sequence in context: A005380 A257557 A124612 * A184697 A124613 A296626

Adjacent sequences:  A230436 A230437 A230438 * A230440 A230441 A230442




Mamuka Jibladze, Nov 04 2013



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Last modified July 22 20:51 EDT 2019. Contains 325226 sequences. (Running on oeis4.)