On the triangle conjecture

Volume 14, Number 5

INFORMATION PROCESSING LETTERS

23 July 1982

ON THE TRIANGLE CONJECTURE

Clelia DE FELICE Istituto di Matematica dell’Wnivetsitb di Napoii, Napoli, Italy Received 15 April 1982; revised version received 13 May 1982 Keywotak Codes, free monoids, triangle conjecture

1. Introduction

such that

S2 ,...,s,

EN

In the following N denotes the set of nonnegative integers, A = {a, b} a two-letters alphabet and A* the free monoid of base A. An element w of A* is called a word and ]w] denotes its length. A subset X of A* is a code over A if X is the base of the free submonoid X* of A*, i.e., if any work of X*\ 1 has a unique factorization in terms of the elements of X. It has been conjectured by Perrin and Schtitzenberger as follows (see [3]).

Xc 6

a*ba”l.

Triangle conjecture. Li;r X c a*ba* be a finite set of words. Set d + 1 = max(lx] I x E X}, if X is a code, then card(X) G d + E.

Condition (ii) of the following lemma has been suggested to me by Prof. A. de Luca.

Some partial results on this conjecture can be found in [ 1, 3, 41. Let us associate to any word a’baj E a*ba* the pair (i, j) E N2. The geometric interpretation suggests the following definitions: If d E N, for any s E N, s ( d, we call row R, (resp. column C,) the subset of a*ba*: R,=(a’ba”IiEN,i
Let us consider the case in which X occupies two rows only. It has been proved in [4]. Proposition 1.1. Let X be a finite subset of a*ba* having at most two rows occupied. If X is a code, then card(X) G d + 1.

2. Some technical lemms

Lemma 2.1. Ler X c a*ba* be a code. For ail i, j, e, r, s, t, p, q. v, uEN: (i) rfa’baj, aibac, a’ba” E X and It - rl = 1 - e[, then a’ba” B X. (ii) If a’baj, a' bae, a’ba”, alba”, a Pba‘J (p - v] = b - el and Iu - q] = ft - r[. then a”b X. Roof. (i) If a’ba” E X with t - c = j -- e th of X*\l, then

(resp.C,=(asbaiIiEN,i6d-s)) Then card(R,) = d + 1 - s. We say that in X two elements are in the same row (resp. column) at distance p, p E N, if there exist i, e, m E N such that le - ml = p and a’ba’, ambai E X (resp. aibae, a’barn E X). We say that X occupies n rows if there are s,, 002@0190/82/0000-0000/$02.75

i=l

a’ba jar ba” = ai ba”st’ba” woutd have two factorizations in terms of the elements of X, which is absurd. Likewise, of t - r =e-j, a”baja’ba” = aiba’a’ba”

0 1982 North-Holland

would be a word of in

X* \ the elements of X. (ii) Jf a’ba”’E X and

1 with

two factorizations

k =C(t - s), such that d + 1 - n = h(t - s) + k, d-t l-s=

u-q=t-r,

p-v=j--e,

23 July 1982

INFORMATkON PROCESSING LETTERS

Volume 14, Number 5

(h - l)(t - s) G k,

d + 1 - t = (h - 2)(t - s) + k.

it would follow that a’baja” ba”a’ba” = a’ba”aPbaQa’ba”E X*, that is, X would not be a code. With a similar argument one proves the other cases. El Lemma 2.2. Let X be a finite subset of a*ba* that occupies at most two rows. X is a code wer A if and only if X satisfies Lemma 2.1(i). Proof. We have to prove the ‘if’ part of the lemma. If X occupies one row, the result is true, so we suppose that X occupies exactly two rows. The proof follows by contradiction. Let us suppose that there is a word of X*\ 1 with two factorizations in terms of the elements of X and of minimal length: = at;baq;at;baq; , . . atkba& where na2, t; = t,, q’ =q and for all iE ti:, +ty+l and q:-+ ti+, = (1 ,***,n- I}, qi#q:, 4i

+

fi+l*

The last equation gives, for i = n - 1, It:,--t,l=lq’,-,

--9,-J;

moreover, since X occupies only two rows, we have that IQ’,-, -%-,I

3. Main result Lemma 3.1. Let R,, R,, R, be three rows such that n c s < t and X c a*ba* a finite code that occupies the rows R,, R,, R t only. Morewer, if in X there are two l?lementsin a same column at distance t - n and two elements in a same row at distance s - n, then card(X) G d + 1.

Proof. Let us consider a map J/ defined in X as follows:

Iq; - q, I

;;;=norp=

{

=§.

t,

Since X is a code and by hypothesis ihere are two elements in a row at distance s - n, by Lemma 2.1(i) one has that a” ba” E X =) aeba” @X, i.e., JI is an injective map. Thus, Y = #(X) has the same cardina.lity as X. By the hypothesis, if in Y there are two elements in a same column at distance t - n, then Y c a*ba* is a finite set that occupies exactly two rows: R, and R t. Moreover, for all e, m E N such that le - ml = t - n one has that (3 . 1)

= 19;--%I*

Then Iti - t nI = hypothesis. Cl

;;;I

\Cl(aebap)=

which contradicts

the

The proof of the following lemma is omitted since it is straightforward.

where p is equal to t or to n. If p = t, this is true since in such a case aebaP E Y implies that aebaP E X, and (3.1) follows by Lemma 2.1(i). Let us then suppose p = n. If aeban E Y, we have e:ither

Lemma 2.3. Let n, s, t, d be four nonnegative a’ba” E X

integers such that

ds2,

n
s-n=t-s.

Then there exist uniqur integers h, k E N, h > 2, 198

(3 .2)

or aebas +EX.

(3 .3)

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23 July 1982

In the case (3.2) by Lemma 2.1(i) one has that amban e X, and by Lemma 2.l(ii) that a”ba” @ X. Thus amban $ Y. In a similar way, in the case (3.3) by Lemma 2.1(i) we have that a*ba” B X and by Lemma 2.l(ii) that amban e X, i.e., amban e Y. Eq. (3.1) implies, by Lemma 2.2, that Y is a code over A that occupies the rows R, and R, only, so that from Proposition 1.l it follows that card(Y) s d + 1. Hence card(X) G d + 1. 0

In each of these cases one has that card(X) G d + 1 - n. In fact, let us suppose, for instance, that n, s, t verify the second case. Thus:

Lemma 3.2. Let R,, R,, R, be three rows such that

and a”ba” E X, then amban, aqba” @ X. Therefore, for u = 0,. . ., k - 1, one has that

R, = (au +i(s-n)ban, ,V+j(S--n)ba” 1 i “0,

l,..., 3a+2;

u=O,

l,...,k-

j =O, l,..., 3a+ 1; v=k,...,s-n-

Ie-mJ=s-n,

le - ql = 2(s - n)

card((auei(s-n)ban

I i=0,...,3a+2)

and, for v=k ,..., s-n-

Proof The proof is similar to that of Lemma 3.1, except that s and t exchange their roles. 0

card(XnR,)c(s-n)(a+

d + 1 - n = (3a + l)(s - n) + k, d+ 1 .-s=3a(s-n)+k, d + 1 - t = (3a - l)(s - n) + k, d + 1 - n = 3a(s - n) + k, d + 1 - s = (3a - l)(s - n) + k, d + 1 - t = (3a - 2)(s - n) + k.

1,

1).

card(XnR,)e(s-n)(a+

l),

card(X n R,) G (s - n)a + k. *

By the foregoing inequalities one has that card(X) = card(X n R,) + card(X f? R,) +card(X

d + 1 - n = (3a + 2;r(s - n) -+ k, d + 1 - s = (3a + l)(s - n) + k, d + 1 - t = 3a(s - n) + k,

l] nX)ga+

Likewise, one has that

XcR,UR,uR,, n
d + 1 - n = 2(s - n) + k, d+ 1 -s=(s-n)+k, d+l-t=k,

+1

hence,

Lemma 3.3. Let X be a finite set such that

an (YE N, a > 1, uniquely determined by h, such that one of the following conditions is verified:

nX)e

1,

card((a’+j(“-“)ba”/j=0,...,3a+

Proof. If (h, k) is the pair of Lemma 2.3, there is

1).

Moreover, by the made hypothesis, if e, m, q are three nonnegative integers such that

n < s < t and X c a*ba* a finite code that occupies the rows R n, R,, R t only. Moreover, [f in X there are two ele’ments in a same column at distance s - n and two elements in a same row at distance t - n, then card(X) G d + 1.

We suppose that in X there are neither two elements in a same row at distance s -.- n nor two elements in a same row at distance t - n = 2(s - n). Then card(X) =Gd + 1 - n.

1;

n R,)

G (3a + 2)(s - n) + k = d + 1 - n. In the other cases, the result follows in a similar way. Cl Proposition 3.4. Let X c a*ba* be a finite code that occupies at most three rows, R ,.,, R,, R t, such that nCs
Case 1. In X there are no pairs of elements in a same column. Case 2. In X there are two elements in a same column at least. In Case 1 the number of elements of X is less then or equal to the number of columns. Then card(X) s d + I - r~ 199

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In Case 2 either (a) there are in X two elements in a same column at distance t - n, or (b) there are in X two elements in a same column at distance s - n. Let us consider Case 2(a). If there are in X two elements in a same row at distance s - n, the result follows by Lemma 3.1; otherwise, by Lemma 1.1(i), in X there are no pairs of elements in a same row at distance t - n and the result follows by Lemma 3.3. Case 2(b) follows in a similar way. Cl Acknowledgment

I am indebted to Prof. A. de Luca for his suggestions and comments.

200

23 July 1982

References G. HanseI, Btionnettes et cardinaux, Discrete Math. 39 (1982) 331-335. I21D. Perrin and M.P. Schtitzenberger, 1Jn probleme &mentaire de la thkrie de l’information, Call. Internat. du CNRS no. 276, Thkwie de YInformation (1977) 249-260. 131 D. Perrin and M.P. Schikxnberger, A conjecture on sets of differences of integer pairs, BuBeth BATCS 5 (1978)27-29; J. Combin. Theory Ser. B 30 (1981) 91-93. PI J.E. Pin and I. Simon, A note on the triangle conjecture, J. Combin. Theory Ser. A 32 (1982) 106-109.

Ill