AbstractLet A be an n-square (0, 1)-matrix, let ri denote the i-th row sum of A, i=1, …, n, and let per (A) denote the permanent of A. Then per(A)≤∏i=1nri+21+2 where equality can occur if and only if there exist permutation matrices P and Q such that PAQ is a direct sum of 1-square and 2-square matrices all of whose entries are 1

Minc, Henryk

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An inequality for permanent of (0, 1)-matrices 

JOURNAL OF COMBINATORIAL THEORY 2, 321-326 (1967) 
An Inequality for Permanents of (0,1)-Matrices* 
HENRYK MINC 
Department of Mathematics, 
University of California, Santa Barbara, California 
Communicated byH. J. Ryser 
ABSTRACT 
Let A be an n-square (0, 1)-matrix, let ri denote the i-th row sum of A, i = 1 ..... n, 
and let per(A) denote the permanent of A. Then 
per(A) ~< H ri q- ~/-2- 
,.1 I+VT 
where equality can occur if and only if there exist permutation matrices P and Q such 
that PAQ is a direct sum of l-square and 2-square matrices all of whose entries are 1. 
I f  A = (ai~) is an n-square matr ix then the permanent of  A is defined 
by 
per(A)  = ~ f i  aio(i). (1) 
aeS n i=1 
An up-to-date survey of  the theory of  permanents was given in [2]. 
Many  propert ies of  the permanent  funct ion are similar to those of the 
determinant. In part icular,  there is the fol lowing analog of  Laplace ex- 
pansion by the k-th co lumn 
per(A)  = ~ aik per(A(i[k)), (2) 
* This work was supported by the Air Force Office of Scientific Research under 
Grant AFOSR 432-63. 
321 
322 MINC 
where A(i[ k) denotes the (n -- 1)-square submatrix of A obtained by 
deleting the i-th row and the k-th column of A. 
Permanents and particularly permanents of (0,1)-matrices are of 
combinatorial significance. Various bounds for permanents have been 
proved and conjectured [1, 2]. In [3] I have conjectured that 
per(A) ~ f i  (r~!)1/% (3) 
i=1 
where r~ = ~j~ ai3 is the i-th row sum of A. I also proved that 
n r~ + 1 
per(A) i=111 2 (4) 
with equality if and only if A is a permutation matrix. In [1] Jurkat and 
Ryser proved that 
per(A) ~ f i _  i=1 ( r i ' )a /n ( r i -k l )  (n - tO/n"  2 " (5) 
which is better than the bound given by (4). In the present paper I prove 
the following improvement of (4). 
THEOREM. I f  A ~- (ai~) is an n-square (0, 1)-matrix then 
per(A) ~ f i  rl + ~2-  (6) 
i=1 I+V~-  
Equafity in (1) occurs if and only if there exist permutation matrices P 
and Q such that PAQ is a direct sum of 1-square and 2-square matrices 
all of whose entries are 1. 
The bounds in (5) and (6) are not comparable. In fact, the inequality 
(5) becomes equality for matrices all of whose entries are 1, while (6) 
is equality in case A is a direct sum of 2-square matrices all of whose 
entries are 1. 
We first prove two combinatorial lemmas. 
LEMMA 1. I f  A z (aij) iS an n-square (0, 1)-matrix without a zero row 
then 
min ~ aij/r~ _< 1 (7) 
i=1 
AN INEQUALITY FOR PERMANENTS OF (0, I)-MATRICES 323 
PROOF : 
Thus 
~ (~ aUrO : ~ 1 ~ aii 
=1 i=1 i=1 ri j=l  
- -  r i 
i=1 r i  
~n .  
min ~ ai/r~ ~ 1. 
j i=l 
Clearly, Lemma 1 implies that we can permute the rows and the 
columns of a (0, 1)-matrix without a zero row so that 
1 ~ 1, (8) 
i=1 r i  
where c is the first column sum and ri the i-th row sum of the permuted 
matrix. Note that both sides of inequality (6) are invariant under per- 
mutations of rows and columns. 
LEMMA 2. Let A ~ (aij) be an n-square (0, 1)-matrix without zero rows 
for which (8) holds. Then 
(%/2-+1)~ 1 _<[*I (1+ 1 ). (9) 
Equality holds in (9) i f  and only if either c = rx = 1, or e ~ r 1 ~-  r 2 = 2. 
PROOF: Let d---- [Ii~l (ri + 21/2 -- 1). Then 
(V/2-+ 1) ~ 1 _ %/2+ 1 ~ f i  ( r t+V-2 -  - 1) 
i=1 ri + %/~-  1 d i=1 t=lt~i 
1 
= -d- {(C 2 + 1)Ec_i + 2Eo_2 
+ ~ t('~/2---1) t 2E~_t} , 
t=3 
(10) 
where Ek denotes the k-th elementary symmetric function of rx, ..., rc ; 
Eo ~ 1. Thus 
324 MINC 
(~/-2-+ I) k 1 
i=1 r i  -JU V ~- -  1 
since 
1 
-7 (('g"2+l)Ec-l+2Ec -2+ ~] "V/2tEc_t) 
t=3 
d -7 t=o 
_ 1 (Ec_ , _Ee)+ 1 f i  ( r ,+~/-2-)  
d -7 i=1 
<[ I  1 1+ 1 
r i+  - 1 
(ll) 
which is nonpositive, by Lemma 1. If either c = rl = 1 or c = rx = r= 
= 2, then clearly (9) is equality. Conversely, if (9) is equality then (11) 
is equality which, together with (10), implies E,_ a = 0, i.e., c < 2. It 
easily follows, by direct computation, that either cl = rl = 1 or e = rl 
=r2=2.  
PROOF OF THE THEOREM: If A has a zero row then (6) is a strict ine- 
quality. Suppose now that A has no zero rows. Since both sides of (6) 
are invariant under permutations of rows and columns of A, we can as- 
sume without loss of generality that 
a l l  = . . . .  ael  = 1, ae+l ,  1 = 9 9 9 = an,  1 ---- 0 
and that (8) holds. We shall prove (6) by induction on n. Assume there- 
fore that (6) holds for all k-square (0, 1)-matrices, k < n. Then 
per(A) = k per(A(ill)) 
i=1 
j r  
= k 1+%/2- ( f i  r~+~v/2 - - -1 )  ( 
~=~ri+V'T-1 J=~ r j+v/5  - 
AN INEQUALITY FOR PERMANENTS OF (0, 1)-MATRICES 325 
But 
i--1 ri _~_ "~/T - -  1 3"-1 rj Au V -2  
(12) 
< 1 
i=1 ri + ~V/-2 - 1 / j=l rj -~ ~/~--- 1 
by Lemma 2. Hence 
per(A) ~ f i  ri + %/2- 
j=l 1 -~- V2-  
If equality holds in (6), the inequality (12) must be equality and thus, 
by Lemma 2, either c = rl = 1 or c = rl =- r~ = 2. In the first case it 
follows immediately from the induction hypothesis that A is of the form 
described in the theorem. If c = rl = r2 = 2 then, by the induction 
hypothesis, both A(lf l) and A(211 ) must be of the required form. It is 
easilyseenthatinthiscasePAQmustbeadirectsumof[ll 11] and an 
(n -- 2)-square (0, 1)-matrix and the result again follows from the in- 
duction hypothesis. Conversely, if PAQ is the direct sum of k 2-square 
and n -- 2k 1-square matrices all of whose entries are 1, then 
i=1 1 + %/2- 
This completes the proof of the theorem. 
Since per(A ~') = per (A) we have 
per (A)<min{f i r~+~v/2- , f i  c i+  /~} 
i=1 1 + %/-2- i=a 1 + 
(13) 
where ci denotes the i-th column sum of A. However, the inequality 
per(A) < f i  si + 
,-11 
where s~ ---- min(ri, ci), i = 1 ..... n, is false. For example, if 
326 MINC [ iy] 
A- -  1 
1 
then per (A)= 4, while 
s, q- ~v/2- r  - 3 + ~r V /~-= 4 V/-~- - 2 < 4. 
i=11 q- V/-2 - 1 +~r  
REFERENCES 
I. W. B. JURKAT AND H. J. RYSER, Matrix Factorizations of Determinants and Per- 
manents, J. Algegra 3 (1966), 1-27. 
2. M. MARCUS AND H. MINC, Permanents, Amer. Math. Monthly 72 (1965), 577-591. 
3. H. MINC, Upper Bounds for Permanents of (0, 1)-matrices, Bull. Amer. Math. Soc. 
69 (1963), 789-791. 


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An inequality for permanent of (0, 1)-matrices

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