login
Number of contractible "tight" meanders of width n.
1

%I #73 Sep 03 2024 01:30:50

%S 1,2,6,14,34,68,150,296,586,1140,2182,4130,7678,14368,26068,48248,

%T 86572,158146,281410,509442,901014,1618544,2852464,5089580,8948694,

%U 15884762,27882762,49291952,86435358,152316976,266907560,469232204,821844316

%N Number of contractible "tight" meanders of width n.

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

%C 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:

%C 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)},

%C 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,

%C 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.

%C 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.

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

%C From _Roger Ford_, Jul 05 2023: (Start)

%C 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.

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

%C e exterior length 1

%C ____________ arches arches

%C / ______ \

%C e1 / / \ \ top = 2 top = 2

%C /\ / / /\1 \ \

%C / \ / / / \ \ \

%C \ \ / / \ \ / / bottom = 2 bottom = 2

%C \ \/1 / \ \/1 / total = 4 total = 4

%C \______/ \______/

%C e e Example T(4,4).

%C (End)

%H Mamuka Jibladze, <a href="/A230439/b230439.txt">Table of n, a(n) for n = 1..100</a> (first 64 terms by Martin Plechsmid)

%H Vincent Coll, Colton Magnant, and Hua Wang, <a href="http://arxiv.org/abs/1206.2705">The Signature of a Meander</a>, arXiv:1206.2705 [math.QA], 2012.

%H Vladimir Dergachev and Alexandre Kirillov, <a href="http://www.heldermann-verlag.de/jlt/jlt10/DERGALAT2E.PDF">Index of Lie algebras of Seaweed Type</a>, J. Lie Theory, 10 (2000), 331-343

%H Mathoverflow, <a href="http://mathoverflow.net/questions/146802/special-meanders">"Special" meanders</a>

%H Dmitri I. Panyushev, <a href="https://www.mathnet.ru/eng/mmj18">Inductive Formulas for the Index of Seaweed Lie Algebras</a>, Moscow Math. J., 1 (2001), 221-241.

%e 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}).

%p # program based on the C code by Martin Plechsmid:

%p proc()

%p local n,a,b,d,r;

%p option remember;

%p if args[1]=1 then

%p 1

%p elif nargs=1 then

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

%p else

%p n:=args[1]; a:=args[2]; b:=args[3];

%p if b=[] then

%p `+`('procname(n,a,[k])'$'k'=1..n)

%p elif a[1]=b[1] then

%p 0

%p elif a[1]<b[1] then

%p procname(n,b,a)

%p else

%p d:=a[1]-b[1];

%p r:=irem(b[1],d);

%p if r>0 then

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

%p else

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

%p fi

%p fi

%p fi

%p end;

%t (* program based on the C code by Martin Plechsmid: *)

%t f[n_,a_,b_]:=Which[

%t n==1, 1,

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

%t a=={} || First[a]<First[b],f[n,b,a],

%t First[a]==First[b], 0,

%t True,

%t f[n-First[b],Join[With[{d=First[a]-First[b]},With[{r=Mod[First[b],d]},If[r==0,{d},{d-r,r}]]],Rest[a]],Rest[b]]];Table[f[n,{},{}],{n,20}]

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

%K nonn

%O 1,2

%A _Mamuka Jibladze_, Nov 04 2013