ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL

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1 ADSS, Volume 3, Number 3, 2013, Pages Aditi International ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL KOKO KALAMBAY KAYIBI Abstract Let i j T( M; = tijx y be the Tutte polynomial of the matroid M and let Q( M, N; = qijklx y u v be the linking polynomial of the matroid pair ( M, N ) In [1] it is proved that the coefficients of the Tutte polynomial of connected matroids are decreasing on minors That is, tij ( M ) tij( M ) if M is a minor of M, where t ij (M ) stands for the ij coefficient of the polynomial T ( M; Moreover, let ( i ) ( ij) if i i and j It is also proved in [2] that if t ij > 0, then t i > 0 for all ( i ) such that ( i ) ( ij) This paper extends these properties to the coefficients of the linking polynomial More specificall we prove that the coefficients of the linking polynomial are decreasing on minors We also prove that if that if q ijkl 0, then q i 0 for all ( i ) such that ( i ) ( ijkl) write We write 1 Introduction A\ B for the set difference between the sets A and B We A\ e or A U e instead of A \{ e} or A U {e}, respectively A matroid M defined on a finite nonempty set E consists of the set E and a collection B of subsets of E, satisfying the following axioms B1: B is not empty B2: If B 1 and B 2 B then, for all e B 2 \ B 1, there is f B 1 \B 2 such that ( B2 U f )\ e B Elements of B are the bases of the matroid M The independent sets of M are all the subsets of the bases A circuit of M is a minimal subset Keywords and phrases: tutte polynomial, matroid function

2 not contained in any basis Let and let KOKO KALAMBAY KAYIBI 46 E 2 denote the set of all the subsets of E + N be the set of non-negative integers The rank function of M, denoted by r, is a function from E + 2 to N, where, for X E, r( X ) is the cardinality of a largest independent set I contained in X Let M be a matroid defined on E with rank function r and let X E The matroid M \ X is the matroid defined on E\ X with rank function r x = r The operation consisting of construction M \ X from M is the deletion operation The matroid with rank function r x defined as follows M X is the matroid defined on E\ X r x ( A) = r( A U X ) r( X ) The operation consisting of constructing contraction operation Both M X from M is the M \ X and M X are minors of M The matroid M * is the matroid defined on E with rank function r * defined as follows r *( X ) = X r( E) + r( E\ X ) An element e is a loop of M if r ( e) = 0 That is, e belongs to no basis of M The element e is a coloop of M if r *( e) = 0 That is, e belongs to every basis of M We say that e is ordinary if e is neither a loop nor a coloop A matroid defined on E is connected if for any two elements e, f E, there is a circuit X such that X contains e and f A matroid is disconnected otherwise Suppose that a matroid M with rank function r is disconnected Then E can be partitionned into E 1 and E 2 such that e E 1 and f E 1 And, there exits two matroids M 1 and M 2 such that M 1 is defined on E 1 with rank function r and M 2 is defined on E 2 with rank function r, and there is no circuit of M containing both e and f The matroid M can thus be written as M = M 1 M 2 For a matroid M defined on E, the Tutte polynomial of M, denoted by T ( M;, is a two-variable polynomial defined as follows

3 ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL 47 T( M; r( E) r( x) x r( x) = ( x 1) ( y 1), (1) x E where r is the rank function of M This polynomial is much researched on since it encodes much information about the matroidal properties of combinatorial structures and is found useful in counting combinatorial invariants For example, it is widely used in Statistical Mechanics where partitions functions of Potts models are evaluations of the Tutte polynomial See [11, 14, 15, 16] for an extensive exposition to this topic An important property of the Tutte polynomial is that it obeys a deletion/contraction recursion as follows (i) T ( M; = T( M \ e; + T( M e ; if e is ordinary (ii) T ( M; = T( M1; T( M2; if M = M 1 M2 (iii) T ( M; = 0 if M consists of a single loop (i T ( M; = 1 if M consists of a single coloop This recursive definition allows to compute the Tutte polynomial of any matroid recursivel by recursive deletion/contraction operations Let M and N be two matroids defined on the set E with rank function r and s respectively We call the pair (M, N) a matroid pair The linking polynomial of (M, N), denoted by Q ( M, N;, is defined in [17] as follows Q( M, N; = ( x 1) x E r( E) r( x) ( y 1) x r( x) s( E) s( X ) X s( X ) ( u 1) ( v 1) (2) The linking polynomial contains the Tutte polynomial of a matroid as a specialization and is equivalent the Tutte polynomial of a matroid perspective defined and studied by Las Vergnas in a series of papers [6, 7, 8] It also partially contains the Tutte invariant of 2-polymatroids defined by Oxley and Whittle in [9, 10]

4 KOKO KALAMBAY KAYIBI 48 2 Results The following is a property of the coefficients of the linking polynomial, which is proved as Proposition 1 in [17] and is used in the sequel Proposition 21 Let Q( M, N; = qijklx y u v If the coefficient q ijkl in Q ( M, N, is non zero then it is positive if and only if i + j + k + l has the same parity as r + s Similarly to the Tutte polynomial of a matroid, the linking polynomial obeys a deletion/contraction recursion given in [17] For the sequel of this article we need only the fact that if and N, then e E is ordinary in at least one of M Q( M, N) = ΘQ( M \ e, N,\ e) + Q( M e, N e) (3) where Θ and are functions of the variables of Q depending on the nature of the element e For example if e is ordinary in M and coloop in N we have that Θ = u 1 and = 1 If e is ordinary in both M and N, then Θ = = 1 For any e E, we call the term Θ Q( M \ e, the deletion term and Q( M e, N e) the contraction term With this definition, we next give a consequence of Proposition 21 which we shall use to prove some more results all Proposition 22 Let (M, N) be a matroid pair defined on E Then for e E which are ordinary in at least one of M and N and all (ijkl), the sign of q ijkl in the deletion and contraction terms are not opposite Proof There are several cases to consider, but we only presente one of them Suppose that ( M, N) has an element e which is ordinary in both M and N Thus r ( E\ e) + s ( E\ e) is even if and only if r ( E e) + s( E e) is even Hence, by Proposition 21, we have that for all ( ijkl), qijkl ( M \ e, is positive if and only if q ijkl ( M e, N e) is positive

5 ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL 49 Let (ijkl) denote elements of the vector space 4 { 0, 1, 2, 3} where i, j, k, l appear as indices of the coefficient q We call (ijkl) a coranknullity sequence Let ijkl q ijkl and q i be two coefficients of Q ( M, N; We write ( ijkl ) ( i ) if i i and j and k and l, and we write ( ijkl ) = ( i ) if i = i and j = and k = and l = We say that (ijkl) and ( i ) are incomparable if some indices in (ijkl) are strictly superior to the corresponding indices in ( i ) and some other indices in (ijkl) are strictly inferior to the corresponding indices in ( i ) Two terms q ijklx y u v and m n p o q mnpox y u v of the Q-polynomial are called like terms if ( ijkl ) = ( mnop) It is a well known fact that if one expands Equation 1 of the Tutte polynomial, some terms, for instance the negative terms cancel out The lack of like terms to cancel out is a property of the Q-polynomial which we give next as Theorem 23 Suppose that the full expansion of equation (2) is given qijklx y u v ijkl sign Then for any (ijkl), all the coefficients of x y u v have the same Proof By induction on the cardinality of E It is routine to check that the proposition holds for all matroid pairs having one element For example Q ( C, C) = ( x 1)( u 1) + 1 = xu x u + 2 Suppose that it holds for all matroid pairs having n elements or less Let ( M, N) be a matroid pair having n + 1 elements There are several cases to consider, but we only present the principal one Suppose that ( M, N) has an element e which is ordinary in both M and N By Equation (3)

6 KOKO KALAMBAY KAYIBI 50 Q ( M, N) = Q( M \ e, + Q( M e, N e) By inductive hypothesis, for all (ijkl), all the coefficients of x y u v have the same sign in both Q ( M \ e, and Q ( M e, N e) Since, by Proposition 22, for all (ijkl), the signs of q ijkl in the deletion term and the contraction term are not opposite, we get that for all (ijkl), all the coefficients of For all x y u v have the same sign in Q ( M, N) X E let corm ( X ) denote the integer r( E) r( X ), nulm( X ) denote X r( X ), corn ( X ) denote the integer s( E) s( X ) and nul N (X ) denote X s(x ) Let ε ijkl denote the family of subsets X E such that corm ( X ) = i, nulm ( X ) = j, corn ( X ) = k, nuln ( X ) = l Let ( M, N) be a matroid pair defined on E We say that X E contributes towards the coefficient q ijkl in the full expansion of Q ( M, N) if X εi for some ( i ) ( ijkl) Using the binomial theorem, one can check that if the contribution of X is given by X E contributes towards the coefficient q ijkl, then i ( i i) + ( j) + ( k) + ( l) ( 1) i j k l Now we can state and prove some corollaries of Theorem 23 Corollary 24 Let ( M, N) be a matroid pair defined on E If ε ijkl is not empty then q ijkl is positive and qijkl = εijkl corm ( Y ) nulm ( Y ) corn ( Y ) nuln ( Y ) + Y ε i j k l ijkl (4) where the sum is only over the subsets Y whose corank-nullity sequence ( i l) > ( ijkl) Proof Suppose ε ijkl is not empty So let X E be in ε ijkl In the expansion of equation (2), the monomial corresponding to X is ( x 1) ( y 1) ( u 1) ( v 1) Multiplying out this product gives the sum

7 ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL 51 i+ j+ k+ l x y u v + + ( 1) Thus every element X of ε ijkl contributes + 1 in q ijkl since by Proposition 23 there is no cancellation out of like terms Hence, qijkl εijkl Now let Y εijkl with cor m ( Y ) = i, nul M ( M ) =, cor N ( Y ) =, nul N ( Y ) = If ( i ) > ( ijkl) then using the binomial theorem, we get that the contribution of Y into q ijkl is ( i i) + ( j) + ( k) + ( l) i ( 1) i j k l 0 since by Proposition 23, there is no cancellation of coefficients Obviousl if ( i l) < ( ijkl) or ( i l) and (ijkl) are incomparable, then at least one of the binomial coefficients is zero Thus corm ( Y ) nulm ( Y ) corn ( Y ) nuln ( Y ) = i j k l 0 Hence the result It is interesting to know in which cases qijkl = εijkl We shall partly answer this question later Corollary 25 If q ijkl is negative then qijkl = Y ε ijkl corm i ( Y ) nulm ( Y j ) corn k ( Y ) nuln ( Y l ) where the sum is only over the subsets Y whose corank-nullity sequence is less than (ijkl) Proof If q ijkl is negative, then by Corollary 24 the set ε ijkl is empty Thus the only contribution to q ijkl comes from the subsets Y such that corm ( Y ) = i, nulm ( Y ) =, corn ( Y ) =, nuln ( Y ) = and ( i ) > (ijkl) For each such Y, the contribution is

8 KOKO KALAMBAY KAYIBI 52 i i j k l which is obviously negative Hence the result Corollary 26 The coefficient q ijkl = 0 if and only if ε i is empty for all ( i ) ( ijkl) Proof Suppose that q ijkl = 0 but ε ijkl is not empty Then by Corollary 24 we get that q ijkl > 0 Contradiction Thus if q ijkl = 0, then ε ijkl is empty Suppose now that q ijkl = 0 but ε i is not empty for some ( i ) > ( ijkl) Let X εi In the full expansion of equation (2) the contribution of X toward q ijkl is ( i i) + ( j) + ( k) + ( l) i ( 1) i j k l which is not zero since i > i, > j, > k, > l And by Proposition 23, no other terms cancel out the contribution of X Hence q ijkl 0 Contradiction Thus if q ijkl = 0 then ε i is empty for all ( i ) ( ijkl) Since Conversel suppose that ε i is empty for all ( i ) ( ijkl) ε i is empty for all ( i ) > ( ijkl), then eventuall in the full expansion of Q ( M, N), only the subsets X εijkl contribute towards q ijkl But ε ijkl is also empty Thus no X E contributes towards q ijkl Thus q ijkl = 0 Corollary 27 If ( ijkl ) ( i ) and q ijkl = 0 then q i = 0 The contra-positive of Corollary 27 is another property of the Tutte polynomial which extends to the Q-polynomial First, we extend the definition of connectedness of matroid as follows A matroid pair (M, N) defined on E is said to be disconnected if there exist two matroid pairs ( M 1, N 1) and ( M 2, N 2 ) such that for some disjoint sets E 1 and E 2 of E

9 ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL 53 such that E 1 U E2 = E, we have that M 1 and N 1 are defined on E 1, M 2 and N 2 are defined on E 2 such that M = M 1 M 2 and N = N 1 N 2 In [2], the set of coefficients of the Tutte of a given matroid M is proved to have a convex like property More formally we have the following Theorem 28 Let M be a connected matroid If ti, j( M ) > 0 then for all ( i ) with ( 00) < ( i ) ( ij) we have that t i, > 0 This extends to the Q-polynomial, without the connectivity requirement Formally we have the following Theorem 29 Let ( M, N) be a matroid pair defined on E Suppose that ( 0000) ( i0 j0k0l0 ) ( i1 j1k1 l1 ) Then for all matroid pairs, if is not zero, then q i1 j1 k1l 1 q i0 j0k0l0 0 Another property of the Tutte polynomial which extends to the Q- polynomial without the connectivity requirement is given in the following result In [1], it is proved that the coefficients of the Tutte polynomial of connected matroids are decreasing on minors More formally we have the following Theorem 210 [[1], Corollary 69] If N is a minor of a connected matroid M, then each coefficient t ij in T ( N; is less than or equal to the corresponding coefficient in T ( M; for all i and j In Theorem 210, the condition that the matroid be connected is necessary Theorem 210 extends to the Q-polynomial without the connectivity requirement This is the object of the following proposition

10 KOKO KALAMBAY KAYIBI 54 First we need some definitions Let (M, N) be a matroid pair defined on a set E and let X and Y be two disjoint subsets of E The matroid pair ( M \ X Y, N \ X Y ) is called a minor of ( M, N) Theorem 211 Let the matroid pair ( M 1, N 1) be a minor of ( M, N) Then, for all (ijkl), qijkl( M1, N1) qijkl( M, N), where a stands for absolute value of a Proof By induction on the cardinality of E It is routine to check that the proposition holds for all the matroid pairs defined on a set of cardinality one or two Suppose now that the proposition holds for all matroid pairs having n elements or less There are many cases to consider, but we only present the fundamental one Suppose that (M, N) has n + 1 elements and an element e which is ordinary in both M and N Then, by Equation (3) we get that Q ( M, N; = Q( M \ e, N\ e; + Q( M e, N e ; Thus for all corank-nullity sequences (ijkl), qijkl ( M, N ) = qijkl( M \ e, + qijkl( M e, N e) (5) If either ( M \ e, or q ijkl ( M e, N e) or both are zero, then by equation (5), the proposition holds by induction If both q ijkl ( M \ e, and q ijkl ( M e, N e) are not zero, then by Proposition 21, we have that q ijkl ( M \ e, is positive if and only if q ijkl ( M e, N e) is positive Thus, considering equation (5) we get that q ijkl ( M, N ) is positive if and only if q ijkl ( M e, N e) is positive Hence qijkl( M, N ) qijkl( M e, N e) and qijkl( M, N ) qijkl( M \ e,

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