A generalization of the matching property of nested sets and of the relevant results is presented. Motzkin words are associated to this notion and are related to non planar permutations.Présentation d'une généralisation de la propriété de couplage entre deux ensembles de paires non croisées et des résultats correspondants. Certains mots de Motzkin sont associés à cette notion, ainsi qu'aux permutations non planaires

Panayotopoulos, Antonios

Tsikouras, Panos

OpenEdition

 Mathématiques et sciences humaines
Mathematics and social sciences 
149 | Printemps 2000
Varia
The multimatching property of nested sets
La propriété de multicouplage d'ensembles de paires non croisées
Antonios Panayotopoulos and Panos Tsikouras
Electronic version
URL: http://journals.openedition.org/msh/2808
DOI: 10.4000/msh.2808
ISSN: 1950-6821
Publisher
Centre d’analyse et de mathématique sociales de l’EHESS
Printed version
Date of publication: 1 March 2000
ISSN: 0987-6936
 
Electronic reference
Antonios Panayotopoulos and Panos Tsikouras, « The multimatching property of nested sets », 
Mathématiques et sciences humaines [Online], 149 | Printemps 2000, Online since 10 February 2006,
connection on 02 May 2019. URL : http://journals.openedition.org/msh/2808  ; DOI : 10.4000/
msh.2808 
© École des hautes études en sciences sociales
23
Math. &  Sci. hum., (38e année, n° 149, 2000, p. 23-30)
THE MULTIMATCHING PROPERTY OF NESTED SETS
Antonios PANAYOTOPOULOS1, Panos TSIKOURAS1
RÉSUMÉ – La propriété de multicouplage d’ensembles de paires non croisées.
Présentation d’une généralisation de la propriété de couplage entre deux ensembles de paires non
croisées et des résultats correspondants. Certains mots de Motzkin sont associés à cette notion, ainsi
qu’aux permutations non planaires.
MOTS-CLÉS!–  Mot de Dyck, Mot de Motzkin, Couplage, Non croisée, Permutation planaire.
SUMMARY – A generalization of the matching property of nested sets and of the relevant results
is presented. Motzkin words are associated to this notion and are related to non planar permutations.
KEYWORDS – Dyck word, Motzkin word, Matching, Nested, Planar permutation.
1. INTRODUCTION
The matching property of nested sets was introduced in [3] and was applied on planar
permutations in [2].
In this paper the matching property is generalized with the introduction of k-matching
nested sets. Four recursive constructions are presented for the generation of k-matching
nested  sets;  every  pair  of  k-matching  nested  sets  can  be  constructed  this  way. The
k-matching nested sets are related to Motzkin words. A Motzkin word is assigned to each
non planar permutation, thus giving a correspondence between non planar permutations
and k-matching nested sets.
k-matching nested sets are represented by non intersecting, closed, plane curves,
creating figures in the shape of level curves (isotherms, indifference curves, etc.).
Furthermore, they could be related to components of electrical circuits (see [5]).
We recall that a set  U  of pairwise disjoint sets of pairs of elements of  Ù*  is called
nested set of pairs if for every  {a,b},{c,d} Œ U  we never have  a < c < b < d.  Let domU
be the set of all the elements of Ù *  that belong to some pair of  U  and  N2 n  the set  of
all  nested  sets  of  pairs  U  with  domU = [2n]. We  say  that  two  nested  sets  U, L  are
matching iff  domU = domL  and  domA = domB,  A Õ U, B Õ L  imply that either  A =
B = ∅  or  domA = domU.
                                                
1 University of Piraeus, Department of Informatics – 80 Karaoli & Dimitriou Street, 18534 Piraeus,
Greece, e-mail!: antonios@unipi.gr.
A. PANAYOTOPOULOS, P. TSIKOURAS24
2. k-MATCHING PROPERTY
Let  U  be a nested set and  B Õ domU;  we call  B  complete if for every  a Œ B  with
{a,b} Œ U,  we have  b Œ B.  We write  U/B = {{a,b} Œ U : a Œ B}.
PROPOSITION 2.1.  Let  U, L  be two nested sets of pairs with  domU = domL.  Then,
there exists a partition  B1, B2, …, Bk  of  domU  with  Bi  complete, such that the sets
U/Bi, L/Bi, i Œ [k]  are matching.
PROOF.  Let  U1 Õ U,  L1 Õ L  with  domU1 = domL1  and  domU1  is minimum but
not!zero (such a pair exists, since  domU = domL).  U1, L1  are matching nested sets.
Since, obviously, the sets  U\U1  and  L\L1  are also nested, we repeat the procedure
for!these sets, thus obtaining the matching nested sets  U2, L2.  We continue recursively
until!we get that  Uk = U \ (U1 » U2 » … » Uk-1),  Lk = L \ (L1 » L2 » … » Lk-1)  are
nested, thus determining uniquely the required partition  {domU1, domU2,…, domUk}  of
domU.          n
We call two nested sets of pairs  k-matching,  k Œ Ù *  in order to indicate the number
of blocks contained in the  k-partition  of the above proposition. In this paper we deal with
k ≥ 2,  since for  k = 1  we get the notion of matching nested sets (see [3]).
We find the blocks of the partition formed by the  k-matching  nested sets  U, L  by the
following recursive procedure.
Suppose  we  have  already  found  the blocks  B1, B2, …, Bm.  We choose the pair
{a,b} Œ U  containing the smallest element  a  of  [2n]  not already included in  B1 » B2 »
… » Bm.  Then we choose the pair  {b,c} Œ L, then the pair  {c,d} Œ U  and so on, until
we choose the pair of  L  containing  a.  The union of these pairs forms the block  Bm+1.
So, the sets U={{1,10}, {2,3}, {4,5}, {6,9}, {7,8}, {11,12}}
L={{1,4}, {2,3}, {5,6}, {7,12}, {8,11}, {9,10}}
are 3-matching.
Indeed we have:
U!: {1,10} {9,6} {5,4} {2,3} {7,8} {11,12}
L!: {10,9} {6,5} {4,1} {3,2} {8,11} {12,7}
thus getting the blocks  B1={1,4,5,6,9,10},  B2={2,3},  B3={7,8,11,12}  and the matching
nested sets:
U/B1={{1,10}, {4,5}, {6,9}} U/B2={{2,3} U/B3={{7,8}, {11,12}}
L/B1={{1,4}, {5,6}, {9,10}} L/B2={{2,3}} L/B3={{7,12}, {8,11}}
(see Figure 1).
THE MULTIMATCHING PROPERTY OF NESTED SETS 25
 
U
 1       2      3     4       5      6       7       8       9      10   11    12
L
Figure 1.  3-matching nested sets
As we have already pointed out, the  k-matching  property is a generalization of the
matching property of nested sets, which has been studied in [3].  So, we present here the
generalization of the relevant propositions, after giving the following definition.
Given  U Œ N2 n  and  c Œ Z,  we define the translation  U + c = {{a + c, b + c} : {a,b}
Œ U},  where all numbers are taken  mod2n,  so that  U + c Œ N2 n.
PROPOSITION 2.2.  If  U, L  are  k-matching,  then  U + c,  L + c  are also  k-
matching.
A  pair  {a,b} Œ U  is  called  outer  pair   if  there  is  no  pair  {c,d} Œ U  such that
c!< a < b < d,  whereas it is called short pair  if there is no  c Œ domU  with  a < c < d.
In  [6]  it has been shown that every element of  N2n+2  may be written in either one or
the other of the forms!:
U* = U » {{2n + 1, 2n + 2}}  and  Uab=(U \ {{a,b}}) » {{a, 2n + 2},{b, 2n+ 1}},
where  U Œ N2 n  and  {a,b}  is an outer pair of U.
PROPOSITION 2.3.  If the sets  L, U  of  N2 n  are  k-matching,  then the sets  L*,  Uab  of
N2n+2  are also  k-matching.
PROPOSITION 2.4.  Let  L, U  be two  k-matching  nested sets of  N2 n.  If  {a,b}  and
{c,d}  are outer pairs of  L  and  U  respectively, that do not belong to the same block of
the  k-partition,  then the sets  Lab,  Ucd  are   (k-1)-matching nested  sets of  N2n+2.
The proofs of propositions 2.2, 2.3 and 2.4 are straightforward extensions of the
respective proofs of [3].
For each nested set  U  of  N2 n  and each outer pair  {a,b}  of  U,  we now define the
nested set of  N2n+2!:
Ua = U1 » (U2 + 1) » {{a, 2n +2}}
where  U1 = U/[a - 1],  U2 = U/[a, 2n]  and  [0] = ∅.  Notice that this partition of  U  into
U1,  U2  always exists, since  {a,b}  is an outer pair.
A. PANAYOTOPOULOS, P. TSIKOURAS26
PROPOSITION 2.5.  If  L, U  are k-matching nested sets of  N2 n  and  {a,b}, {a,c}  are
outer pairs of  L, U  respectively then  La,  Ua  are  (k+1)-matching nested sets of  N2n+2.
PROPOSITION 2.6.  If  L, U  are  k-matching  nested  sets  of  N2 n  then  L » {2n + 1,
2n + 2}, U » {2n + 1, 2n + 2}  are  (k+1)-matching nested sets of  N2n+2.
Propositions  2.3,  2.4,  2.5  and  2.6  suggest  four  constructions  for the generation of
k-matching nested sets of N2n+2,  using  h-matching  nested sets of  N2 n,  with  h ≤ k.
The generalization of the result of [3] for the converse procedure is given in the
following proposition.
PROPOSITION 2.7.  Every pair  U, L  of k-matching nested sets of  N2n+2,  may be
generated by  h-matching nested sets of  N2 n,  with  h ≤ k.
Let  U  be a nested set of  N2 n. We define  
˜ U   to be the nested set such that {a,b} Œ U
if and only if  {2n + 1 - b, 2n + 1 - a} Œ ˜ U .
PROPOSITION 2.8.  U, L  are k-matching nested sets if and only if  ˜ U , ˜ L   are k-
matching nested sets.
3. MOTZKIN WORDS
We recall that a word  w Œ {x, x }*  is called Dyck if  wx=w x   and for every
factorization  w = uv  of  w,  we have  u x ≥ u x   where  wx,ux (resp.
w
x 
,u
x 
)  is the number of occurrrences of the letter  x  (resp. x )  in  w, u.
A word  w Œ {x, x ,y, y }*  is called Motzkin if the word obtained by deleting all
occurrences of  y, y  from  w,  is a Dyck word.
Now,  to  each  pair  of  k-matching  nested  sets  U, L Œ N2 n  we  assign  a  word  w!=
z1 z2 … z2 n  of  {x, x ,y, y }*  with!:
x,   if   i < j, h
x ,   if   h, j < i
zi =
y,   if  h < i < j
y ,   if  j < i < h
where  {i,j} Œ U,  {i,h} Œ L.
THE MULTIMATCHING PROPERTY OF NESTED SETS 27
So, from our example of section 2, we get the following word
w = x    x    x    y    y    y    x    y    y    x    y   x 
which is a Motzkin word.
In general, we have the following proposition.
PROPOSITION 3.1.  The word  w  is a Motzkin word, having the same number of
occurrences of  y  and  y .
PROOF.  There is a Dyck word, and hence a set of left and right parentheses,
corresponding to each nested set  U  and  L  (see [1]).  Let  X  be the subset of  [2n]
corresponding to the positions of the word  w,  where both parentheses are left; similarly
X'   for both right parentheses,  Y  for left  U  parenthesis and right  L  parenthesis and
finally  Y'   for right  U  parenthesis and left  L  parenthesis. Notice that if in the  i-th
position both parentheses are left then  i < j, k  and so  zi=x; hence,  X  equals the
number of occurrences of  x  in  w.  Similarly  X' ,Y  and  Y'   equal the number
of occurrences of  x , y, y  respectively.
Let  Xp  (resp. X p
'
, Yp , Yp
' )  be the subset of  X  (resp. X' ,Y, Y' )  that we get if we deal
with the first  p  pairs of parentheses only. Since  U  corresponds to a Dyck word we have
that  Xp»Yp≥X p
'
» Yp
' ,  since  Xp»Yp  (resp. X p
'
» Yp
' )  refers to the positions
where the  U  parentheses are left (resp. right).  Since  Xp,Yp  and  X p
'
, Yp
'   are obviously
disjoint, we get that  Xp + Yp ≥ X p
'  + Yp
' ,  i.e.
Xp-X p
'  ≥ Yp
'  - Yp (1)
From  L  we  similarly get  Xp»Yp
'  ≥ X p
' »Yp  and hence, finally
Xp - X p
'  ≥ Yp - Yp
'  (2)
From  (1), (2)  we get that  Xp - X p
'  ≥ Yp - Yp
'  ≥ 0
i.e. Xp ≥ X p
'  (3)
From the definition of Dyck words we also have that
X + Y = X'  + Y'   and   X + Y'  = X'  + Y
getting X = X'  (4)
and Y = Y'  (5)
(3)  and  (4)  prove that the word which we get if we delete all occurrences of  y  and  y 
from  w  is a Dyck word  (and so  w  is a Motzkin word), whereas (5) proves that the
number of occurrences of  y  and  y  in  w  is the same.      n
We also have the following result.
A. PANAYOTOPOULOS, P. TSIKOURAS28
PROPOSITION 3.2.  Each digram  x x , x y , y x , y y  (resp. x x , xy, y x , y y)  in  w,
corresponds to a short pair of  U  (resp.  L).
So, applying recursively the above proposition, we get from the Motzkin word  w  of
our example the nested sets  U  and  L  as follows!:
w  =   x    x    x    y    y    y    x    y    y    x      y     x 
 1    2    3    4     5     6    7    8     9    10     11    12
 x    y    y    x 
 1    6    9    10
 x    x 
 1    10
So,   U={{1,10},{2,3},{4,5},{6,9},{7,8},{11,12}.
Similarly, for  L  we have!:
w  =   x    x    x    y    y    y    x    y    y    x     y     x 
 1    2    3    4     5    6     7    8     9    10    11   12
 x    y    x    y    y    x 
 1    4    7    8    11   12
 x     x 
 7    12
So,   L={{1,4},{2,3},{5,6},{7,12},{8,11},{9,10}}.
We define the level of a pair  {a,b}  of a nested set  U  recursively, as follows:
• {a,b}  is of level  1  if it is a short pair.
• {a,b}  is of level  k + 1  if every pair  {c,d}  with  a < c < b  is of level less or equal to
k  and there is a least one such pair of level k.
Obviously, each pair of maximum level in  U  is an outer pair.
Notice that the above procedure determines not only the pairs of each nested set, but
their level too: the pairs using the elements of the block  Bi,  i Œ [k]  of the corresponding
partition of proposition 2.1, have level  i.
So, in our previous example, the pairs  {2,3},{4,5},{7,8},{11,12}  of  U  are short pairs,
whereas  {6,9}  is of level 2 and  {1,10}  is of level 3.
4. NON PLANAR PERMUTATIONS
Planar permutations were defined by Rosenstiehl in [4]. A permutation  s  on  [2n]  with
s(1) = 1  is called planar  (p.p.)  if the sets  Us={{s (2i - 1), s (2i)} : i Œ [n]},
THE MULTIMATCHING PROPERTY OF NESTED SETS 29
Ls = {{s(2i), s(2i + 1)} : i Œ [n - 1]} » {{s(2n), 1}}  are both nested;
e.g.  s = 1   6   7   8   5   4   9   10   11   12   3   2   is a p.p. with
Us = {{1,6},{2,3},{4,5},{7,8},{9,10},{11,12}},
Ls = {{1,2},{3,12},{4,9},{5,8},{6,7},{10,11}};  (see Figure 2).
     Us
1 2 3 4 5 6 7 8 9 10 11 12
     Ls
Figure 2.  The nested sets  Us, Ls  of a  p.p.  s
Let  P2 n  be  the  set  of  all  p.p.’s  of  lenght  2n.  It  is  easy  to  check  that  for every
s Œ P2 n  we have that  s(i)  is odd iff  i  is odd.
Propositions 3.4 and 3.5 of [3] give a correspondence between pairs of matching
nested sets and p.p.’s on [2n]. This result is generalized by the following proposition:
PROPOSITION 4.1.  To each non planar permutation  s  of length  2n  corresponds a
pair of  k-matching nested sets.
Indeed, as we have already developed a procedure giving the two matching nested sets
of pairs from the word  w = w1 w2…w2 n  of  {x, x , y, y }*,  it is enough to assign a
particular word  w  to each non planar permutation  s = s(1) s (2)…s(2n).
We do this as follows:
x,     if     s(i - 1) > s(i) < s(i + 1)
x ,     if     s(i - 1) < s(i) > s(i + 1)
y,     if     s(i - 1) < s(i) < s(i + 1)     and   i   is odd, or
ws(i) =
    s(i - 1) > s(i) > s(i + 1)     and   i   is even
y ,     if     s(i - 1) < s(i) < s(i + 1)    and   i   is even, or
    s(i - 1) > s(i) > s(i + 1)     and   i   is odd
for every  i Œ [2n]  (where the indices are taken  mod2n).
e.g.  For the non planar permutation  s = 1   3   2   5   6   9   12   11   7   10   8   4  we get
w = x   x   x   y   y   y   x   y   y   x   y   x  giving the 3-matching nested sets  U, L  of
the example of section 2.
A. PANAYOTOPOULOS, P. TSIKOURAS30
It is easy to establish a correspondence between the 1-1 mappings of length  2n  to a
subset of Ù *,  and the set  P2 n  of  p.p.’s  on  [2n];  indeed, if  t = t(1) t (2)…t(2n)  is
such a mapping, we can consider the corresponding  s = s(1) s (2) … s (2n)  of  P2 n to
be the unique  p.p. for which  s(i) < s(j)  if and only if  t(i) < t(j),  i, j Œ [2n].
e.g. If  t = 1   6   5   4   9   10, then the corresponding  s Œ P6  is  s = 1   4   3   2   5   6.
The matching nested sets  U/Bi,  L/Bi,  i = 1, 2, …, k  of proposition 2.1 give rise to  1-1
mappings of length  2ni  to subsets of  Ù *.
So, from proposition 4.1  we get the following corollary.
COROLLARY 4.2.  To each non-planar permutation corresponds a family of p.p.’s.
e.g.  We have already seen that from  s = 1  3  2  5  6  9  12  11  7  10  8  4  we get  the  3-
matching nested sets  U,L  which, in turn, give  t1 = 1   10   9   6   5   4,  t2 = 2   3   and  t3
= 7   8   11   12, giving the corresponding p.p.’s   s1 = 1  6  5  4  3  2,  s2 = 1   2   and  s3
= 1   2   3   4.
REFERENCES
[1] KNUTH, D. E., The Art of Computer Programming, vol.1, Addison Wesley, 1982.
[2] PANAYOTOPOULOS, A., «Generating planar permutations», Journal of
Information and Optimization Sciences, 18(2), 1997, 281-287.
[3] PANAYOTOPOULOS, A., SAPOUNAKIS, A., and TSIKOURAS, P., «The
matching property of nested sets», Australas. Journal of Combin., 15, 1997, 17-24.
[4] ROSENTIEHL, P., «Planar permutations defined by two intersecting Jordan curves»,
Graph Theory and Combinatorics, Academic Press, 1984, 259-271.
[5] ROSENSTIEHL, P., «Grammaires acycliques de zigzags du plan», Mots, Hermès,
1990, 367-383.
[6] SAPOUNAKIS, A., and TSIKOURAS, P., «Nested sets of pairs», Journal of
Information and Optimization Sciences, 16(3), 1995, 549-555.


The multimatching property of nested sets

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