aa r X i v : . [ m a t h . C O ] S e p SYZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS
JOSEPH P.S. KUNGA
BSTRACT . It follows from a theorem of H. Derksen [
J. Algebraic Combin.,
30 (2009)43–86] that the Tutte polynomial of a rank- r matroid on an n -set is “naturally” a linearcombination of Tutte polynomials of rank- r size- n freedom matroids. However, the Tuttepolynomials of rank- r size- n freedom matroids are not linearly independent. We constructtwo natural bases for these polynomials and as a corollary, we prove that the Tutte poly-nomials of rank- r matroids of size- n spans a subspace of dimension r ( n − r ) + 1 . Wealso find a generating set for the linear relations between Tutte polynomials of freedommatroids. This generating set is indexed by a pair of intervals, one of size and one of size , in the weak order of freedom matroids. This weak order is a distributive lattice and asublattice of Young’s partition lattice. Primary 05B35; Secondary 05B20, 05C35, 05D99, 06C10, 51M04, 52B401. T
WO MATROID INVARIANTS
We begin with the cadet, the G -invariant, introduced by Derksen [4] in 2009. Let M bean ( n, r ) -matroid, that is, a rank- r matroid on the set { , , . . . , n } with rank function rk and closure cl . For a permutation π on { , , . . . , n } , the rank sequence r ( π ) of π is thesequence r r . . . r n defined by r = rk( { π (1) } ) and for j ≥ ,r j = rk( { π (1) , π (2) , . . . , π ( j ) } ) − rk( { π (1) , π (2) , . . . , π ( j − } ) . It is immediate that r j = 0 or , there are exactly r ’s, and the set { π ( j ) : r j = 1 } is abasis of M. A bitsequenceis a sequence of zeros and ones, and an ( n, r ) -sequenceis a bit sequenceof length n with (exactly) r ’s. Let [ r ] be a variable or formal symbol, one for each ( n, r ) -sequence r, and G ( n, r ) be the vector space of dimension (cid:0) nr (cid:1) consisting of all formallinear combination of symbols [ r ] with coefficients in a field K of characteristic . The G -invariant G ( M ) and its coefficients g r ( M ) are defined by G ( M ) = X π [ r ( π )] = X r g r ( M )[ r ] , where the first sum ranges over all n ! permutations of { , , . . . , n } . A specialization ofthe G -invariant taking values in an abelian group A is a function assigning a value in A to each symbol [ r ] . The G -invariant is fundamental because by a theorem of Derksen andFink [5], it is a universal valuative invariant on matroid base polytopes, in the sense thatevery valuative invariant on base polytopes is a specialization of the G -invariant.The veteran is the Tutte polynomial. It is a classical and well-studied object. To clarifynotation, we recall the definition of the Tutte polynomial: for a rank- r matroid M on a set S with n elements, the Tutte polynomial T ( M ) and its coefficients t ij ( M ) are defined by T ( M ) = T ( M ; x, y ) = X i,j ≥ t ij ( M ) x i y j = X A ⊆ S ( x − r − rk( A ) ( y − | A |− rk( A ) . We denote by T ( n, r ) the subspace in the algebra K [ x, y ] of polynomials in the variables x and y with coefficients in the field K spanned by the Tutte polynomials of ( n, r ) -matroids.We assume a basic acquaintance with the theory of Tutte polynomials (see [2, 6] for sur-veys).Derksen [4] showed that there is a specialization sending the G -invariant to the Tuttepolynomial. The specialization is given explicitly in the following lemma. Lemma 1.1.
The assignment G ( n, r ) → K [ x, y ] , [ r r . . . r n ] n X m =0 ( x − r − wt( r r ...r m ) ( y − m − wt( r r ...r m ) m !( n − m )! , where the Hamming weight wt( r r . . . r m ) is the number of ’s in the initial segment r r . . . r m , sends the G -invariant of a matroid to its Tutte polynomial. Explicitly, T ( M ) = 1 n ! X π n X m =0 (cid:18) nk (cid:19) ( x − r − rk( { π (1) ,π (2) ,...,π ( m ) } ) ( y − m − rk( { π (1) ,π (2) ,...,π ( m ) } ) ! . This specialization extends to a linear transformation Sp : G ( n, r ) → K [ x, y ] . Ourobjective in this paper is to determine the kernel of Sp , show that its image is T ( n, r ) , anddescribe natural bases for T ( n, r ) . To do this, we need a partial order on ( n, r ) -sequences.This order is described in Section 2. With this order, we describe in Section 3 the syzygiesof Sp , that is, linear combinations in ker Sp . Freedom matroids are introduced in Section4. The G -invariants of freedom matroids form a natural basis of G ( n, r ) . Hence the Tuttepolynomials of freedom matroids span T ( n, r ); however, they fail to form a basis. Wedescribe two subsets of freedom matroids whose Tutte polynomials form a basis in Section5 and a generating set for linear relations on Tutte polynomials of freedom matroids inSection 6. In Section 7, we use a basis found in Section 5 to give another proof of a theoremof Brylawski [1] describng a basis for linear relations on coefficients of Tutte polynomials.The last two sections are computational. We give formulas for Tutte polynomials in one ofthe bases found in Section 5 and as an example, compute explicitly the freedom matroidsand their relations in T (5 , .
2. A
PARTIAL ORDER ON RANK SEQUENCES
Let S ( n, r ) be the set of ( n, r ) -sequences. We define the (partial) order D in the follow-ing way. If r and s are two ( n, r ) -sequences, then s D r if for every index j, ≤ j ≤ n,s + s + · · · + s j ≥ r + r + · · · + r j , in other words, reading from the left, there are always at least as many ’s in s as thereare in r. Using the notation where a stands for a sequence of a (consecutive) ’s and b asequence of b ’s, this order has maximum r n − r and minimum n − r r . The partial order ( S ( n, r ) , D ) is a sublattice of Young’s (partition) lattice (see, for example, [11, p. 288]).We shall use D as the underlying order for “straightening” or Gr¨obner basis arguments. Inparticular, no esoteric properties of Young’s lattice will be used. YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 3
An intuitive way to think of the order D is to view a sequence r as a lattice path fromthe origin (0 , to the corner ( r, n − r ) where a is a north step and a is an east step.Then s D r if and only if as lattice paths, r never goes higher than s. The lattice paths lieinside the rectangle with opposite corners (0 , and ( r, n − r ) . Tilting the rectangle so thatit pirouettes on the corner (0 , , we can also think of a sequence r as an order ideal on thedirect product of a r -chain and an ( n − r ) -chain, with D equal to set-containment ⊆ . Asorder ideals are subsets closed under intersections and unions, ( S ( n, r ) , D ) is a distributivelattice with meet equal to intersection and join equal to union.The sequence r has an ascent at position i if i ≥ , r i − = 0 , and r i = 1 . It has a descent at i if i ≥ , r i − = 1 , and r i = 0 . In ( S ( n, r ) , D ) , s covers r if s = r r , r = r r . An element j in a lattice is a join-irreducible if j covers at most one element. In the lattice ( S ( n, r ) , D ) , a sequence is join-irreducible if and only if it has at most one descent. Inparticular, join-irreducibles are sequences of the form a b c d , with a + b + c + d = n and b + d = r. Working upside-down, an element m in a lattice is a meet-irreducible if m covers at most one element. A sequence is meet-irreducible if and only if it has at most oneascent and meet-irreducibles are sequences of the form a b c d , with a + b + c + d = n and a + c = r. We denote the set of join-irreducibles (respectively, meet-irreducibles) in ( S ( n, r ) , D ) by J (respectively, M ). A simple counting argument gives | J | = r ( n − r ) + 1 = | M | . By definition, g r ( M ) ≥ . The support supp( M ) of an ( n, r ) -matroid M is the set { r : g r ( M ) > } . Lemma 2.1.
The support of an ( n, r ) -matroid M is an order filter in ( S ( n, r ) , D ) . Proof.
It suffices to show that if s covers r, and r ∈ supp( M ) , then s ∈ supp( M ) . To dothis, let r = r r , and s = r r , where r has length λ. As g r ( M ) > , it is the ranksequence of a permutation i i . . . i n (in one-line notation). Then s is the rank sequence ofthe permutation i i . . . i λ i λ +2 i λ +1 . . . i n and we conclude that g s ( M ) ≥ . (cid:3)
3. S
YZYGIES FOR ker Sp The cornerstone of our theory is the following lemma.
Lemma 3.1.
Let r be a ( λ, ρ ) -sequence and s a bit sequence such that r s is an ( n, r ) -sequence. Then Sp ([ r s ] − [ r s ]) = ( x − r − ρ − ( x + y − xy )( y − λ − ρ − ( λ + 1)!( n − λ − . Proof.
We use Lemma 1.1, noting that the summands in Sp ([ r s ]) and Sp ([ r s ]) are thesame except at m = λ + 1 . When m = λ + 1 , the summands for Sp ([ r s ]) and Sp ([ r s ]) are ( x − r − ρ − ( y − λ − ρ ( λ + 1)!( n − λ − x − r − ρ ( y − λ − ρ +1 ( λ + 1)!( n − λ − and the lemma follows. (cid:3) An interval I of height in ( S ( n, r ) , D ) has the form shown in Figure 1, where r r r is an ( n, r ) -sequence. We associate with I the linear combination sz ( I ) de-fined by sz ( I ) = [ r r r ] − [ r r r ] − [ r r r ] + [ r r r ] . JOSEPH P.S. KUNG r r r r r r r r r r r r F IGURE
1. Intervals of height . Lemma 3.2.
For every height- interval I in ( S ( n, r ) , D ) , the linear combination sz ( I ) isin the kernel of Sp . Proof.
By Lemma 3.1, the differences Sp ([ r s ] − [ r s ]) on “opposite” sides of theheight- interval I cancel and Sp ( sz ( I )) = 0 . (cid:3) We note that since ( S ( n, r ) , D ) is distributive, the sublattice generated by the atoms inan interval of height k is a Boolean algebra and give rise to a linear combination with k terms analogous to the -term linear combinations sz ( I ) . The larger linear combinationsare also in ker Sp . An easy argument shows that they are linear combinations of -termlinear combinations sz ( I ) . Let K be the linear subspace in G ( n, r ) spanned by the linear combinations sz ( I ) . If r is meet-reducible (that is, not meet-irreducible), then it has two or more ascents andis the minimum of a height- interval. Hence, [ r ] can be written, modulo K , as a linearcombination with integer coefficients of symbols [ s ] , where s ⊲ r. Repeating this argument,we conclude that in the quotient G ( n, r ) / K , every symbol [ r ] can be written as an integrallinear combination of symbols of meet-irreducibles, that is, [ r ] = X m D r, m ∈ M α m [ m ] mod K , (3 . where the coefficients α m are integers. Working in the opposite way, one derives analogousassertions for join-reducibles and irreducibles. Lemma 3.3.
The symbols [ m ] , m ∈ M , span the quotient G ( n, r ) / K . The symbols, [ j ] , j ∈ J , span G ( n, r ) / K . In particular, dim G ( n, r ) / K ≤ r ( n − r ) + 1 . We end this section with another consequence of Lemma 3.1
Proposition 3.4.
Let M and N be ( n, r ) -matroids. Then x + y − xy divides the difference T ( M ) − T ( N ) of their Tutte polynomials.Proof. By Lemma 3.1, if r covers s, or r is covered by s in ( S ( n, r ) , D ) , then x + y − xy divides Sp ([ r ] − [ s ]) . Since the Hasse or covering diagram of ( S ( n, r ) , D ) is connected, x + y − xy divides Sp ([ r ] − [ s ]) for any pair of ( n, r ) -sequences. Now observe thatthe difference G ( M ) − G ( N ) is a sum of differences [ r ] − [ s ] . From this, we concludethat T ( M ) − T ( N ) is a sum of differences Sp ([ r ] − [ s ]) and hence, x + y − xy divides T ( M ) − T ( N ) . (cid:3) Proposition 3.4 implies that for points ( α, β ) on the curve x + y − xy = 0 , the value of T ( M ; α, β ) is constant on ( n, r ) -matroids M. In fact, if M is an ( n, r ) -matroid and α = 1 ,T ( M ; α, β ) = α n ( α − r − n . YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 5 , F IGURE
2. The freedom matroid F (101010000) .
4. F
REEDOM MATROIDS
Let s be an ( n, r ) -sequence and b , b , . . . , b r be the positions where ’s occur in s, arranged so that b < b < · · · < b r . The freedom matroid F ( s ) with definingsequence s is the matroid on the set { , , . . . , n } in which(1) the elements , , . . . , b − are loops (that is, in the closure cl( ∅ ) ),(2) for ≤ j ≤ r − , b j is added as an isthmus and the elements b j , b j + 1 , b j +2 , . . . , b j +1 − are freely positioned in cl( { b , b , . . . , b j } ) , and(3) b r is added as an isthmus and b r , b r + 1 , . . . , n are freely positioned in the entirematroid.The freedom matroid F ( s ) has a distinguished flag (or maximal chain) of flats X ⊂ X ⊂· · · ⊂ X r , where X i = { , , . . . , b i +1 − } for ≤ j ≤ r − , and X r = { , , . . . , n } . Freedom matroids were first defined by Crapo in [3]; they have been rediscovered manytimes and are also known as nested, counting, or Schubert matroids.The following lemma is immediate from the definition.
Lemma 4.1. (a) Let Y be a rank- i flat in F ( s ) . Then | Y | ≤ | X i | . (b) A set B in { , , . . . , n } is a basis of the freedom matroid F ( s ) if and only if the ( n, r ) -sequence r, defined by r j = 1 if j ∈ B, satisfies s D r. By Lemma 4.1(b), if s D s , then every basis of F ( s ) is a basis of F ( s ); in otherwords, F ( s ) ≥ w F ( s ) , where ≥ w is the weak order on ( n, r ) -matroids. Corollary 4.2.
The weak order on rank- r freedom matroids on { , , . . . , n } is isomorphicto ( S ( n, r ) , D ) . We turn now to G -invariants of freedom matroids. Lemma 4.3. (a) If g r ( F ( s )) = 0 , then r D s. (b) The coefficient g s ( F ( s )) is non-zero.Proof. Let r be the rank sequence associated with the permutation π and b , b , . . . , b r bethe elements of { j : r j = 1 } arranged in increasing order. Then r defines a flag Y ⊂ Y ⊂· · · ⊂ Y r , where Y i = cl( { π (1) , π (2) , . . . , π ( b i ) } ) . By Lemma 4.1(a), b i ≤ | Y i | ≤ | X i | . This implies that r D s. To prove part (b), note that g s ( F ( s )) = ( b − b − b )!( b − b )! · · · ( b r − b r − )!( n − b r + 1)!Φ , where Φ is the number of flags Y ⊂ Y ⊂ · · · ⊂ Y r in F ( s ) such that | Y i | = | X i | for all i. (cid:3) It follows from Lemma 4.3 that the system of equations X r g r ( F ( s ))[ r ] = G ( F ( s )) (4 . JOSEPH P.S. KUNG is triangular with non-zero diagonal coefficients. Thus, we can invert the system (with atriangular matrix) and write a symbol as a linear combination of G -invariants of freedommatroids.For example, when n = 4 and r = 2 , the matrix (cid:0) g r ( F ( s )) (cid:1) and its inverse are , −
12 12 −
13 16 − − −
14 14 − −
124 18 18 −
524 124 , where the rows/columns are indexed by the symbols/freedom matroids with defining se-quences , , , , , . Theorem 4.4.
The G -invariants G ( F ( r )) , r ∈ S ( n, r ) , form a basis for the vector space G ( n, r ) . The change-of-basis matrices between the symbol basis and the G -invariant basisare triangular. In particular, the G -invariants G ( M ) , where M is an ( n, r ) -matroid, span G ( n, r ) and T ( n, r ) equals the image of Sp . Proposition 4.5. If G ( F ( r )) occurs in the expansion of G ( M ) with non-zero coefficient,then r ∈ supp( M ) . Proof.
We use the theory of incidence algebras on partially ordered sets (see [9, 10]).Since g r ( F ( s )) = 0 only if s D r, the entries of the matrix (cid:0) g r ( F ( s )) (cid:1) form an incidencefunction on the partially ordered set ( S ( n, r ) , D ) . Hence, by incidence-algebra theory, theentries of the inverse matrix, which is the change-of-basis matrix ∆ from the symbol basisto the freedom-matroid basis, form an incidence function, that is, the ( s, r ) -entry of ∆ isnon-zero only if s D r. Let ~M be the vector ( g r ( M )) of coefficients of G ( M ) in the symbol basis, Then ∆ ~M is the vector of coefficients of G ( M ) in the freedom-matroid basis. As the entries of ∆ form an incidence function, the coefficient of G ( F ( r )) is non-zero only if r D s for somesequence s in supp( M ) , and by Lemma 2.1, only if r ∈ supp( M ) . (cid:3) The linear map Sp sends G ( M ) to T ( M ) . Since G ( M ) can be written (uniquely) as alinear combination of G -invariants of freedom matroids, we have the following corollary. Corollary 4.6.
The Tutte polynomial of an ( n, r ) -matroid is a linear combination (notnecessarily unique) of the Tutte polynomials T ( F ( r )) , r ∈ S ( n, r ) of freedom matroids. We end with a deletion-contraction lemma.
Lemma 4.7.
Let r r be an ( n, r ) -sequence with a descent (indicated by a ˇ ) at position i. Then T ( F ( r r )) = T ( F ( r r )) + T ( F ( r r )) . Proof.
Since a descent occurs at i, the element i is neither a loop nor an isthmus. Hence,by the deletion-contraction recursion for Tutte polynomials, T ( F ( r r )) = T ( F ( r )) \ i + T ( F ( r )) /i = T ( F ( r r )) + T ( F ( r r )) . (cid:3) YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 7
5. T
WO BASES FOR T ( n, r ) In this section, we find two bases for T ( n, r ) , one coming from the join-irreducibles,the other from the meet-irreducibles of ( S ( n, r ) , D ) . We begin with the basis built fromthe join-irreducibles.
Theorem 5.1.
The Tutte polynomials T ( F (0 a b c d )) , where a + b + c + d = n and b + d = r, are linearly independent in T ( n, r ) . We begin the proof with two formulas. The freedom matroid F (1 r n − r ) is the uniformmatroid U r,n . Writing r = b and n − r = c, its Tutte polynomial is given by T ( U b,b + c ) = b − X j =0 (cid:18) c − jj (cid:19) x b − j + c − X k =0 (cid:18) b − kk (cid:19) y c − k . (5 . For example, T ( U , ) = x + 4 x + 10 x + 20 x + 35 x + 35 y + 15 y + 5 y + y . The freedom matroids F (0 a b c d ) are direct sums of the uniform matroid U b,b + c with a loops and d isthmuses and hence, T ( F (0 a b c d )) = y a T ( U b,b + c ) x d . (5 . For example, T ( F (0 )) = y x + 2 y x + 3 y x + 3 y x + y x . Using these formulas, we write down the matrix Γ of coefficients of Tutte polynomi-als T ( F (0 a b c d )) . The columns of Γ are indexed by freedom matroids with definingsequences r n − r , r − n − r , . . . , n − r n − r − , n − r n − r − , r n − r − , r − n − r − , . . . , n − r − r − , n − r − r − , ... n − r − r , n − r − r − , . . . , n − r − n − r − , n − r − r − , n − r r in the given order and the rows are indexed by monomials x r , x r − , . . . , x , x,yx r , yx r − , . . . , yx , yx,y x r , y x r − , . . . , y x , y x, ... y n − r − x r , y n − r − x r − , . . . , y n − r − x , y n − r − x,y n − r x r , in the given order, with the remaining monomials following in any order. Note that thecolumns and the first r ( n − r ) + 1 rows are divided into n − r blocks , each with r indices,and one additional index. We label the blocks on the rows by the number of ’s at thebeginning of the defining sequence and the blocks on the columns by the exponent of the JOSEPH P.S. KUNG variable y ; in both cases, the label ranges from to n − r − . For example, when r = 3 and n = 5 , there are two blocks, each of size , and Γ is the × matrix x x x yx yx yx y x y y y x y x It follows from formulas (5.1) and (5.2) that in the Tutte polynomial T ( F (0 a b c d )) , all the monomial in blocks to a − have zero coefficient, and in block a, only the mono-mials y a x r , y a x r − , . . . , y a x r − b +1 have non-zero coefficients. In addition, the last Tuttepolynomial, T ( F (0 n − r r )) , equals y n − r x r and the monomial y n − r x r does not occur(with non-zero coefficient) in any other Tutte polynomials T ( F (0 a b c d )) . Lemma 5.2.
The block diagonal submatrix U i , with rows and columns indexed by the i th block, are upper triangular matrices with non-zero diagonal entries. The upper blocksubmatrices, with rows indexed by the i th block and columns indexed by the j th block,with i < j, are zero matrices. The only non-zero entry in the ( r ( n − r ) + 1) st row and ( r ( n − r ) + 1) st column is the diagonal entry. Lemma 5.2 implies that the matrix Γ restricted to the first r ( n − r ) + 1 rows has theform U . . . ∗ U ∗ ∗ U . . . ... ∗ ∗ ∗ U n − r −
00 0 0 . . . , where the block diagonal submatrices U i are r × r upper triangular matrices and the aster-isks ∗ are r × r matrices. From this, it is evident that the matrix Γ has rank r ( n − r ) + 1 . We conclude that dim T ( n, r ) ≥ r ( n − r ) + 1 . Combining this with Corollary 3.3 and the first homomorphism theorem for vector spaces,we obtain r ( n − r ) + 1 ≥ dim G ( n, r ) / K ≥ dim T ( n, r ) ≥ r ( n − r ) + 1 , and hence K = ker Sp and G ( n, r ) / K ∼ = T ( n, r ) . The following results are immediate consequences.
Theorem 5.3.
The linear combinations sz ( I ) , where I ranges over all height- inter-vals of ( S ( n, r ) , D ) , span ker Sp . The symbols [ j ] , j ∈ J , form a basis for the quotient G ( n, r ) / ker Sp and the symbols [ m ] , m ∈ M , form a basis for G ( n, r ) / ker Sp . YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 9
Theorem 5.4.
The Tutte polynomials T ( F ( j )) , j ∈ J , form a basis for T ( n, r ) . In partic-ular, dim T ( n, r ) = r ( n − r ) + 1 . We call { T ( F ( j )) : j ∈ J } the join-irreduciblebasis of T ( n, r ) . We will now consider the basis built from the meet-irreducibles, that is, ( n, r ) -sequencesof the form a b c d . When c = 1 , F (1 r − b n − r − b ) is the paving matroid on { , , . . . , n } with one non-trivial copoint { , , . . . , r − b } . Freedom matroids defined by meet-irreducibles can be characterized by their cyclic flats. Recall that a set is cyclic if it is aunion of circuits.
Lemma 5.5.
Let n > r. An ( n, r ) -matroid M is isomorphic to a freedom matroid F ( s ) , where s is a meet-irreducible, if and only if M contains exactly one cyclic flat, or exactlytwo cyclic flats, one of which is the entire set { , , . . . , n } . Proof. If a = r, then c = 0 and F (1 a b c d ) ∼ = U r,n . If a < r and d = 0 , then F (1 a b c d ) is the direct sum of the uniform matroid U a,a + b and r − a isthmuses. In thegeneral case, when a < r and d > , F (1 a b c d ) has two cyclic flats, { , , . . . , a + b } and { , , . . . , n } . This argument can be reversed and the lemma follows. (cid:3)
In contrast to the Tutte polynomials in the join-irreducible basis, the Tutte polynomials T ( F (1 a b c d )) have complicated formulas. These formulas will be described in Section8. The next lemma gives a formula for the case b = c = 1 . Lemma 5.6.
Let r ≥ and n ≥ . Then T ( F (1 r − n − r − )) = T ( F (1 r n − r )) − ( x + y − xy ) . Proof.
A simple calculation yields G ( F (1 r n − r )) = (cid:18) nr (cid:19) r !( n − r )![1 r n − r ] , G ( F (1 r − n − r − )) = (cid:18)(cid:18) nr (cid:19) − (cid:19) r !( n − r )![1 r n − r ] , + r !( n − r )![1 r − n − r − ] and hence G ( F (1 r n − r )) − G ( F (1 r − n − r − )) = r !( n − r )!([1 r n − r ] − [1 r − n − r − ]) . We can now finish the proof by applying Sp and Lemma 3.1. (cid:3) Lemma 5.6 can also be proved using induction and deletion-contraction. The methodof proof yields a more general result, which we state without a proof. (For the definitionof circuit-hyperplane relaxation, see [8, p. 39].)
Proposition 5.7.
Let M ′ be obtained from M by a circuit-hyperplane relaxation. Then G ( M ′ ) − G ( M ) = r !( n − r )!([1 r n − r ] − [1 r − n − r − ]) ,T ( M ′ ) − T ( M ) = x + y − xy. To show that { T ( F ( m )) : m ∈ M } is a basis, we first show that the G -invariants G ( F ( m )) , m ∈ M form a basis of the quotient vector space G ( n, r ) / ker Sp . To see this, we use equations (3.1) and (4.1) to obtain, for m ∈ M , G ( F ( m )) = X r D m g r ( F ( m ))[ r ]= X r D m g r ( F ( m )) X m ′ D r α m ′ [ m ′ ] = X m ′ D m h m ′ ( F ( m ))[ m ′ ] in the quotient G ( n, r ) / ker Sp . Note that h m ( F ( m )) = g m ( F ( m )) and hence h m ( F ( m )) =0 . Applying Sp , we obtain X m ′ D m h m ′ ( F ( m )) Sp [ m ′ ] = T ( F ( m )) , (5 . a triangular system of equations with non-zero diagonal coefficients relating the sets { T ( F ( m )) : m ∈ M } and { Sp [ m ] : m ∈ M } in T ( n, r ) . However, Theorem 5.3 implies that { Sp [ m ] : m ∈ M } is a basis for T ( n, r ) . Hence, we obtain the following theorem.
Theorem 5.8.
The Tutte polynomials T ( F ( m )) , where m ∈ M , form a basis for T ( n, r ) . We call { T ( F ( m )) : m ∈ M } the meet-irreduciblebasis of T ( n, r ) .
6. B
UILDING LINEAR RELATIONS
In hindsight, the key to finding linear relations on Tutte polynomials of freedom ma-troids is Lemma 5.6. This lemma says that the difference T ( F (1 r n − r )) − T ( F (1 r − n − r − )) equals x + y − xy, a polynomial not depending on r and n. We will show that similar as-sertions hold for linear combinations derived from height- intervals in ( S ( n, r ) , D ) . Thenext lemma gives the smallest case.
Lemma 6.1. T ( F (1010)) − T ( F (1001)) − T ( F (0110)) + T ( F (0101)) = x + y − xy. (6 . Proof.
By Lemma 4.7, T ( F (1010)) = T ( F (101)) + T ( F (100)) ,T ( F (1001)) = T ( F (101)) + T ( F (001)) ,T ( F (0110)) = T ( F (011)) + T ( F (010)) ,T ( F (0101)) = T ( F (011)) + T ( F (001)) . Hence, the left-hand side of equation (6.1) equals T ( F (100)) − T ( F (010)) , which in turn,equals ( x + y + y ) − ( xy + y ) by direct computation. (cid:3) We introduce a compact notation for -term linear combinations. Define L ( r k r k r ) = T ( F ( r r r )) − T ( F ( r r r )) − T ( F ( r r r ))+ T ( F ( r r r )) . Proposition 6.2. L (0 a b k c d k e f ) = x f y a ( x + y − xy ) . YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 11
Proof.
We begin by observing that F (0 a u f ) is the direct sum of F ( u ) , a loops, and f isthmuses and hence, T ( F (0 a u f )) = x f y a T ( F ( u )) and L (0 a b k c d k e f ) = x f y a L (1 b k c d k e ) . Thus, it suffices to prove L (1 b k c d k e ) = x + y − xy. We first show that value of L (1 b k c d k e ) is independent of d. To do this, we use Lemma4.7 at the descent indicated by a ˇ to obtain the deletion-contraction identities T ( F (1 b c d e )) = T ( F (1 b c +1 d +1 e )) + T ( F (1 b c d − e )) ,T ( F (1 b c d ˇ010 e )) = T ( F (1 b c +1 d +1 e )) + T ( F (1 b c d − e )) ,T ( F (1 b c d e )) = T ( F (1 b c d +1 e )) + T ( F (1 b c d − e )) ,T ( F (1 b c d ˇ010 e )) = T ( F (1 b c d +1 e )) + T ( F (1 b c d − e )) . These identities imply that L (1 b k c d k e ) = L (1 b k c d − k e ) . By induction, L (1 b k c d k e ) = L (1 b k c k e ) . We deal next with c. We use the deletion-contraction recursions (at the element indicatedby a ˇ ) T ( F (1 b c e )) = T ( F (1 b c e )) + T ( F (1 b +1 c + e +2 )) ,T ( F (1 b c e )) = T ( F (1 b c e )) + T ( F (1 b c +2 e )) ,T ( F (1 b c e )) = T ( F (1 b c e )) + T ( F (1 b c + e +1 )) ,T ( F (1 b c − e )) = T ( F (1 b c e )) + T ( F (1 b c +2 e )) . to conclude that L (1 b k c k e ) = T ( F (1 b +1 c + e +2 )) − T ( F (1 b c + e +1 )) . By Lemma 5.6, L (1 b k c k e ) = x + y − xy, completing the proof. (cid:3) Noting that a bit sequence has no descent if and only if it equals a b , Proposition 6.2allows us to calculate L ( r k r k r ) when none of the bit sequences r , r , r has a descent.We shall describe a reductionwhich writes L ( r k r k r ) as a linear combination of -termlinear combinations L ( s k s k s ) , where s , s , or s , have fewer descents. To keep trackof the reductions, we use a binary tree.Let u be a bit sequence. A descenttreeof u is a binary tree constructed in the followingway. Start with the sequence u. If u has a descent, say u = u u , then add two de-scendants u u and u u to the node u. Continue for each bit sequence in the partiallyconstructed tree until all the leaves are bit sequences with no descents. There are manydescent trees, one for each ordering by which we choose the descents.By Lemma 4.7, each branching of a descent tree gives a deletion-contraction decom-position of a freedom matroid. Hence, a descent tree of u gives a decomposition of thefreedom matroid F ( u ) into a linear combination of direct sums of loops and isthmuses in F IGURE
3. A descent tree of the leaves yield the Tutte polyno-mial x y + xy + xy + y + y of F (10101) . the Tutte-Grothendieck ring of matroids (see [1, p. 243] or [2, Section 6.2, p. 124]). Thisyields the following lemma. Lemma 6.3.
The multiset of leaves of a descent tree of a bit sequence u depends only on u. Under the specialization a b x b y a , the sum over the multiset of leaves in a descenttree of u equals the Tutte polynomial T ( F ( u ); x, y ) . We define the following functions: f ( u ) = T ( F ( u ); 1 , y ) , g ( u ) = T ( F ( u ); x, , τ ( u ) = T ( F ( u ); 1 , . Theorem 6.4. L ( r k r k r ) = f ( r ) τ ( r ) g ( r )( x + y − xy ) . Proof.
We use the reduction process. Suppose r = r r . Then by deletion-contraction, T ( F ( r r r )) = T ( F ( r r r r )) + T ( F ( r r u r )) . Similar equations hold for the other three Tutte polynomials in the linear combination L ( r k r k r ) and combining these equations, we obtain L ( r k r k r ) = L ( r r k r k r ) + L ( r r k r k r ) . Iterating the reduction on r , and repeating the entire process on r and r , we obtain L ( r k r k r ) = X ( ℓ ,ℓ ,ℓ ) L ( ℓ k ℓ k ℓ ) , the sum ranging over all triples ( ℓ , ℓ , ℓ ) , with ℓ i a leaf in a descent tree of r i . To finish,we apply Lemmas 6.2 and 6.3. (cid:3)
The next lemma generalizes Lemma 5.6 and can be proved by the method in the proofof Theorem 6.4.
Theorem 6.5.
Let a, b ≥ . Then T ( F ( r a b r )) − T ( F ( r a − b − r )) = f ( r ) g ( r )( x + y − xy ) . Theorems 6.4 and 6.5 give the building blocks for making linear relations on Tuttepolynomials of freedom matroids. The smallest linear relation, holding in T (4 , , is T ( F (1100)) − T ( F (1010)) = x + y − xy = T ( F (1010)) − T ( F (1001)) − T ( F (0110)) + T ( F (0101)) . A more complicated example is obtained by considering the two overlapping height- intervals in Figure 4. The lower height- interval gives L ( r k r k r ) = f ( r ) τ ( r g ( r )( x + y − xy ) YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 13 r r r r r r r r r r r r r r r r r r F IGURE interval gives L ( r k r k r ) = f ( r ) τ ( r ) g (0 r )( x + y − xy ) . Noting that g (0 r ) = g ( r ) , this yields τ L ( r k r k r ) = τ ′ L ( r k r k r ) , where τ = τ ( r ) and τ ′ = τ ( r . Explicitly, τ T ( F ( r r r )) − ( τ + τ ′ ) T ( F ( r r r )) + τ ′ T ( F ( r r r )) − τ T ( F ( r r r )) + ( τ + τ ′ ) T ( F ( r r r )) − τ ′ T ( F ( r r r )) = 0 . A generating set for all linear relations is the set L ( n, r ) consisting of the linear relations τ ( r )[ T ( F ( r a +2 b +2 r )) − T ( F ( r a +1 b +1 r ))] = T ( F ( r r r )) − T ( F ( r r r )) − T ( F ( r r r )) + T ( F ( r r r )) , where r , r , r are bit sequences such that r r r is an ( n, r ) -sequence, a is thenumber of ’s in r , and b is the number of ’s in r . The linear relations in L ( n, r ) areindexed by two intervals in ( S ( n, r ) , D ) : the height- interval [ r r r , r r r ] and lying above it, the height- interval [ r a +1 b +1 r , r a +2 b +2 r ] . The two inter-vals are disjoint or intersect at one sequence. As an example, the pair of intervals indexingthe relation T ( F (11000)) − T ( F (10100)) = T ( F (10010)) − T ( F (10001)) − T ( F (01010))+ T ( F (01001)) is shown in Figure 5. Theorem 6.6.
The set L ( n, r ) is a generating or spanning set for linear relations or syzy-gies on Tutte polynomials of rank- r size- n freedom matroids.Proof. Propositions 6.4 and 6.5 imply that the linear relations in L ( n, r ) hold. To showthat L ( n, r ) is a generating set, note that if s is meet-reducible, then it is the minimum ofa height- interval. Using one of the linear relations in L ( n, r ) , we can write T ( F ( s )) as alinear combination with integer coefficients of Tutte polynomials T ( F ( r )) , r ⊲ s. Repeat-ing this, we can write T ( F ( r )) as an integral linear combination with integer coefficientsof elements in the meet-irreducible basis. We conclude that L ( n, r ) is a generating set. (cid:3) From the proof of Theorem 6.6, we obtain the following proposition.
Proposition 6.7.
The Tutte polynomial of a freedom matroid F ( s ) , s ∈ S ( n, r ) , is a linearcombination with integer coefficients of elements T ( F ( m )) in the meet-irreducible basissuch that m D s. The next proposition follows from Propositions 4.5 and 6.7. F IGURE Proposition 6.8.
Let M be an ( n, r ) -matroid. Then the Tutte polynomial T ( M ) is a linearcombination of Tutte polynomials T ( F ( m )) where m ∈ M ∩ supp( M ) . The girth (or spark) of a matroid M is the minimum size of a circuit in M. Let T k ( n, r ) be the subspace of T ( n, r ) spanned by the Tutte polynomials of ( n, r ) -matroids of girth atleast k. Corollary 6.9.
The subspace T k ( n, r ) has as basis the freedom matroids T ( F ( m )) , m ∈ M and m starts with k − ’s. In particular, dim T k ( n, r ) = ( n − r )( r − k + 1) + 1 . Proof.
Let U be the interval [1 k − n − r r − k +1 , r n − r ] in ( S ( n, r ) , D ) . It is immediatethat the following are equivalent for an ( n, r ) -matroid M :(1) M has girth at least k ; (2) for all ( n, r ) -sequences r, g r ( M ) = 0 implies that r starts with k − ’s;(3) supp( M ) ⊆ U. The map that removes the initial segment k − from each bit sequence in U is an order-isomorphism sending U onto ( S ( n − k +1 , r − k +1) , D ) . From the fact that a bit sequenceis a meet-irreducible if and only if it equals a b c d , it follows that the meet-irreduciblesin U map bijectively onto the meet-irreducibles in S ( n − k + 1 , r − k + 1) and hence,there are ( n − r )( r − k + 1) + 1 such meet-irreducibles. Since the meet-irreducibles in the upper interval U are meet-irreducibles in ( S ( n, r ) , D ) , it follows from Corollary 6.8 thatthe Tutte polynomials T ( F ( m )) , where m is a meet-irreducible in U, form an independentset spanning T k ( n, r ) . (cid:3) The third assertion in the next result uses the easy fact that a rank- r matroid is paving ifand only if it has girth r or r + 1 . Corollary 6.10.
The Tutte polynomials of loopless ( n, r ) -matroids span a subspace ofdimension ( n − r )( r −
1) + 1 , the Tutte polynomials of simple ( n, r ) -matroids span asubspace of dimension ( n − r )( r −
2) + 1 , and the Tutte polynomials of paving ( n, r ) -matroids span a subspace of dimension n − r + 1 .
7. L
INEAR RELATIONS ON COEFFICIENTS OF T UTTE POLYNOMIALS
There are ( r +1)( n − r +1) possible non-zero coefficients t ij , ≤ i ≤ r, ≤ j ≤ n − r, in the Tutte polynomial of an ( n, r ) -matroid. Since dim T ( n, r ) = r ( n − r ) + 1 , thereare n linearly independent linear relations on the coefficients. A set of such relations wasfound by Brylawski [1, Section 6]. For m ≥ , let J m = m X α =0 α X β =0 ( − β (cid:18) αβ (cid:19) t m − α,β . YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 15
One might visualize the linear combinations J m as “tableaux with staircase shape”. Forexample, J = t , J = t − t + t , J = t − t + t + t − t + t J = t − t +3 t − t + t − t + t + t − t + t , J = t − t +6 t − t + t + t − t +3 t − t + t − t + t + t − t + t . The hook H m ( d, a ) is the linear combination in J m defined by H m ( d, a ) = m − d − a X j =1 (cid:18) m − d − ja (cid:19) t j + d,a + (cid:18) m − da (cid:19) t da + m − d − a X k =1 ( − k (cid:18) m − da + k (cid:19) t d,a + k . For example, H (0 ,
0) = t + t + t + t + t − t + 6 t − t + t ,H (0 ,
1) = t + 2 t + 3 t + 4 t − t + 4 t − t ,H (2 ,
1) = t + 2 t − t . The centered hook ¯ H m ( d, a ) is defined by subtracting ( d, a ) from each pair of subscriptsin H m ( d, a ) , that is, ¯ H m ( d, a ) = m − d X j =1 (cid:18) m − d − ja (cid:19) t j + (cid:18) m − da (cid:19) t + m − d X k =1 ( − k (cid:18) m − da + k (cid:19) t ,k . The next theorem is Theorem 6.6 in Brylawski [1].
Theorem 7.1. If n ≥ and ≤ m < n, then the coefficients of the Tutte polynomial of amatroid on a set of size n satisfy J m = 0 . Brylawski proved Theorem 7.1 by showing that when ≤ m < n, J m is a linearcombination of the number of flats (or closed sets) of corank and nullity strictly less than n − r in an ( n, r ) -matroid, and because no such flats exist, J m = 0 . This proof is somewhatcomplicated.We give another proof using the fact that J m = 0 for all polynomials in T ( n, r ) if andonly if J m = 0 for every polynomial in a basis of T ( n, r ) . Thus, to prove Theorem 7.1, itsuffices to check J m = 0 for all polynomials in the join-irreducible basis, that is, the Tuttepolynomials F (0 a b c d ) . Recall that T ( F (0 a b c d )) = x d y a T ( U b,b + c ) and hence, t jk ( F (0 a b c d )) = t j − d,k − a ( U c,c + d ) . Next, note that the coefficients in t ij ( F (0 a b c d )) are non-zero if and only if i = d or j = a but not both. Hence, to check that these coefficients satisfy J m = 0 , it suffices tocheck that they satisfy H m ( d, a ) = 0 . But T ( F (0 a b c d )) satisfy H m ( d, a ) = 0 if andonly if ( ∗ ) T ( U b,b + c ) satisfy the centered hook relation ¯ H m ( d, a ) = 0 . To finish our proof, we will prove assertion ( ∗ ) by induction. We start the induction bychecking that ( ∗ ) holds for the Tutte polynomials T ( U c,c +1 ) and T ( U c,c +2 ) . This is aneasy calculation and we omit the details. For the induction step, we note that by deletion-contraction, T ( U b,b + c ) = T ( U b,b + c − ) + T ( U b − ,b − c ) . By the induction hypothesis, the two Tutte polynomials on the right satisfy ¯ H m ( d, a ) = 0 and by linearity, T ( U b,b + c ) also satisfies ¯ H m ( d, a ) = 0 . This completes our proof ofTheorem 7.1.As J k involves coefficients, such as t k , not in J k − , the linear relations J m = 0 arelinearly independent. Corollary 7.2.
For n ≥ , the linear relations J m = 0 , m = 0 , , . . . , n − , give a basisfor all linear relations on coefficients of Tutte polynomials of ( n, r ) -matroids.
8. F
ORMULAS FOR T UTTE POLYNOMIALS OF MEET - IRREDUCIBLE SEQUENCES
In this section, we calculate the Tutte polynomials T ( F (1 a b c d )) . Theorem 8.1.
When a + c = r and b + c = n − r, then the Tutte polynomial T ( F (1 a b c d )) equals T ( U r,n ) − ( x + y − xy ) b − X j =0 (cid:18) a + jj (cid:19) y j T ◦ ( U c,c + b − j + d ) , where T ◦ ( U c,c + d ) = T ◦ ( U c,c + d ; x ) = T ( U c,c + d ; x, x = c − X k =0 (cid:18) d − kj (cid:19) x c − − k . Proof.
Let D ( a, b, c, d ) = T ( F (1 a + c b + d )) − T ( F (1 a b c d )) . Lemma 8.2.
The differences D ( a, b, c, d ) satisfy the boundary condition D ( a, , c, d ) = 0 and the recursion D ( a, b, c, d ) = D ( a − , b, c, d ) + D ( a, b − , c, d ) . Proof.
The boundary condition is immediate from the definition. To prove the recursion,note that by Lemma 4.7, T ( F (1 a + c b + d )) = T ( F (1 a − c b + d )) + T ( F (1 a + c b − d )) ,T ( F (1 a b c d )) = T ( F (1 a − b c d )) + T ( F (1 a b − c d )) . (cid:3) We now apply Lemma 8.2, bearing in mind that the recursion for D ( a, b, c, d ) on a, b isthe same as the recursion for the binomial coefficients (cid:0) ab (cid:1) , to obtain the following lemma. Lemma 8.3. D ( a, b, c, d ) = b − X i =0 (cid:18) a − ii (cid:19) D (0 , b − i, c, d ) . To use Lemma 8.3, we need an explicit formula for D (0 , b, c, d ) . This can be done usingformulas for Tutte polynomials of uniform matroids.
YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 17
Lemma 8.4. D (0 , b, c, d ) = ( x + y − xy ) b − X j =0 y j T ◦ ( U c,c + b − j + d ) , where T ◦ ( U c,c + d ) = T ◦ ( U c,c + d ; x ) = T ( U c,c + d ; x, x = c − X k =0 (cid:18) d − kk (cid:19) x c − − k . Proof.
We use the fact that D (0 , b, c, d ) = T ( F (1 c b + d )) − T ( F (0 b c d ))= T ( U c,c + b + d ) − y b T ( U c,c + d ) . Thus, the case b = 1 is T ( U c,c + d ) − yT ( U c,c + d − ) = ( x + y − xy ) T ◦ ( U c,c + d ; x ) . (8 . This formula can be proved by a simple calculation using Pascal’s identity for binomialcoefficients. Iterating formula (8.1) and “telescoping” yield Lemma 8.4. For example, T ( U , ) − yT ( U , ) = x + 5 x + 15 x + 35 x + 35 y + 20 y + 10 y + 4 y + y − ( x y + 4 x y + 10 x y + 20 xy + 20 y + 10 y + 4 y + y )= ( x + y − xy )( x + 5 x + 15 x + 35)= ( x + y − xy ) T ◦ ( U , ) and T ( U , ) − y T ( U , )= (cid:0) T ( U , ) − yT ( U , ) (cid:1) + (cid:0) yT ( U , ) − y T ( U , ) (cid:1) + (cid:0) y T ( U , ) − y T ( U , ) (cid:1) = ( x + y − xy )( T ◦ ( U , ) + yT ◦ ( U , ) + y T ◦ ( U , )) . (cid:3) We can now finish the proof of Theorem 8.1 by combining the two lemmas, changingthe order of summation, and using an elementary binomial-coefficient identity: D ( a, b, c, d ) = b − X i =0 (cid:18) a − ii (cid:19) ( x + y − xy ) b − X j =0 y j T ◦ ( U c,c + b − j + d ) = ( x + y − xy ) b − X j =0 (cid:18) a + jj (cid:19) y j T ◦ ( U c,c + b − j + d ) . (cid:3) We note the following special case of Theorem 8.1.
Corollary 8.5. T ( F (1 r − b n − r − )) = T ( U r,n − r ) − ( x + y − xy ) b − X j =0 (cid:18) r − jj (cid:19) y b − − j . From Corollary 8.5, we can easily derive a formula for the Tutte polynomial of a paving ( n, r ) -matroid. F IGURE ( S (5 , , D ) Proposition 8.6.
Let P be a paving ( n, r ) -matroid with f ( s ) copoints of size s. Then T ( P ) = T ( U r,n ) − ( x + y − xy ) X b ≥ f ( r − b ) b − X j =0 (cid:18) r − jj (cid:19) y b − − j . Proposition 8.7.
Let M be an ( n, r ) -matroid having girth at least k. Then t αβ ( M ) = 0 unless ( α, β ) ∈ { ( k,
0) : 1 ≤ k ≤ r } ∪ { ( i, j ) : 0 ≤ i ≤ r − k + 1 , ≤ j ≤ n − r } . In particular, the linear relations J m = 0 , ≤ m ≤ n − , and t ij = 0 , r − k + 2 ≤ i ≤ r and 1 ≤ j ≤ n − r form a basis for all linear relations on the coefficients of (Tutte) polynomials in T k ( n, r ) . Proof.
By Corollary 6.9, T ( M ) is a linear combination of Tutte polynomials T (1 a b c d ) , where a ≥ k − . Thus, it suffices to prove Proposition 8.7 for the freedom matroids F (1 a b c d ) , a ≥ k − . But if a ≥ k − , then c ≤ r − k + 1 and T ◦ ( U c,c + e ; x ) hasdegree at most r − k. Hence, by Theorem 8.1, the coefficients of T (1 a b c d ) satisfy thecondition given in the proposition. (cid:3)
9. C
LASSES OF MATROIDS
In this paper, we focused on the vector spaces G ( n, r ) and T ( n, r ) spanned by the G -invariants or Tutte polynomials of all ( n, r ) -matroids. One could study the vector spaces G ( C ; n, r ) (respectively, T ( C ; n, r ) ) spanned by the G -invariants (respectively, Tutte poly-nomials) of ( n, r ) -matroids in a class C of matroids. When C contains all freedom ma-troids, then the results in this paper hold for G ( C ; n, r ) and T ( C ; n, r ); in particular, theyhold when C is the class of transversal matroids, gammoids, or matroids representable overan infinite field. For classes of matroids not containing all freedom matroids, then almostnothing is known. For example, when C is the class of graphic matroids or the class ofbinary matroids, determining dim G ( C ; n, r ) and dim T ( C ; n, r ) are interesting open prob-lems.10. A PPENDIX . T
UTTE POLYNOMIALS OF RANK - FREEDOM MATROIDS ON ELEMENTS .There are ten rank- size- freedom matroids. Their Tutte polynomials are given by YZYGIES ON TUTTE POLYNOMIALS OF FREEDOM MATROIDS 19 T ( F (11100)) = x + 2 x + 3 x + 3 y + y T ( F (11010)) = x + 2 x + 2 x + xy + 2 y + y T ( F (11001)) = x + 2 x + 2 xy + xy T ( F (10110)) = x + x + x + xy + x y + y + y T ( F (10101)) = x + x + xy + x y + xy T ( F (10011)) = x + x y + x y T ( F (01110)) = yx + yx + yx + y T ( F (01101)) = yx + yx + y xT ( F (01011)) = yx + y x T ( F (00111)) = y x . These polynomials span the vector space T (5 , of dimension . Thus, the space of syzy-gies or linear relations has dimension . The following three linear relations give a basisfor the space of syzygies: T ( F (10101)) − T ( F (11001)) − T ( F (10110)) + 2 T ( F (11010)) − T ( F (11100)) = 0 ,T ( F (01101)) − T ( F (01110)) − T ( F (10101)) + T ( F (10110)) + T ( F (11010)) − T ( F (11100)) = 0 ,T ( F (01011)) − T ( F (01101)) − T ( F (10011)) + 2 T ( F (10101)) − T ( F (11001)) = 0 . Using these relations, we can express the three Tutte polynomials T ( F ( r )) , where r ismeet-reducible, in the meet-irreducible basis. Explicitly, we have T ( F (10101)) = T ( F (11001)) + T ( F (10110)) − T ( F (11010)) + T ( F (11100)) ,T ( F (01101)) = T ( F (11001)) + T ( F (01110)) − T ( F (11010)) + 2 T ( F (11100)) ,T ( F (01011)) = T ( F (01110)) + T ( F (10011)) − T ( F (10110)) + T ( F (11010)) . R EFERENCES[1] T.H. Brylawski, A decomposition theory for combinatorial geometry, Trans. Amer. Math. Soc. 171 (1972)235–282.[2] T. Brylawski, J.G. Oxley, The Tutte polynomial and its applications.
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