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

1,2

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<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.

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)

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.

Example

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}).

Maple

# The 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]<b[1] then
    procname(n, b, a)
   else
    d:=a[1]-b[1];
    r:=irem(b[1], d);
    if r>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

(* Mathematica program based on the C code by Martin Plechsmid *)
f[n_, a___, b___] := f[n, a, b] =
  Which[
   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],
   True,
   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]]
    ]
   ]

Crossrefs

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

Sequence in context: A309536 A257557 A124612 * A184697 A124613 A296626

Adjacent sequences: A230436 A230437 A230438 * A230440 A230441 A230442

Keyword

nonn

Author

Mamuka Jibladze, Nov 04 2013

Status

approved