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The t h r e e border polynomials d e f i n e d above a l l have t h e same corners a. A Let 0.
Let of aij S be a corner of SmB M. If Thus. But 6. We a l s o remark t h a t arguments could be based on the Poincard polynomial. Combining 5.
Example 5. Note 2 is 2s. For example, of 5. M3 three-point c i r c u i t s. S' W e parametrize Lemma 5. The identify for f r,i,s can be found in the papers by Edmonds, Murty, and Young which introduced perfect matroid designs and []. This comes from counting,in two different ways, the i-tuples of independent points much as was done for atoms in 5. That u x,y depends only on ranks appears in [66] and The first formula above for p r,s of the identity: appears in [19],and a simplification gives t h e second formula.
Further, in this case,the parameters of M are recoverable from k j f 1 9 1 K 2. If M is a near-design with parameters If the parameters of a near-design Poincar6 polynomial of where, for a l l 5.
The A-polynomial coefficient, p. A , The degree of a nonzero p. Here, u i s t h e exponent of il il p A of A-degree n-1, u and il coefficient c; ,n-l 1 i n t h e unique polynomial 1fi1, t h e number of atoms, i s t h e of t h e l e a d i n g term of t h i s polynomial corresponding t o t h e number of c h a r a c t e r i s t i c polynomials summed. F M appear on t h e border.
Idy 1. Exercises 5. Extend Example 5. Research Problems 5. What relationships other than 5.
Find examples of near-designs without loops or mltiple points which are not designs or paving matroids. Can the Tutte polynomial of a matroid be reconstructed from the Tutte polynomials of its hyperplanes? We have s e e n many examples 3. Proposition 6.
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Thus, there are and distinct polynomials. Although the estimate above for the number of distinct Tutte polynomials is small compared with the number of matroids, the actual number of Tutte polynomials is even smaller.
Aside from those subtle considerations pointed out in 6. The identities can be characterized completely with reasonable ease,while the search for inequalities leads into many areas of matroid theory and more general mathematics,and seems endless. We make these restrictions,since most applications involve combinatorial geometries no multiple points ,while 4.
Let G be the class of all isthmus-free K. Further, Proposition 6. The dimension of kd K,n equals for the relations which define M K. A basis follow: 2.
The following identities form a basis for the relations which hold -g the coefficients with tunanegative subscripts in the T t l t t e polynolial Ib. M K,n Thus, by 3. Identities a-e a r e obvious. Let Then, K'. However, t h i s can be sharpened. Let M be a connected matroid, and assume t h a t i n i t s T u t t e polynomial. To determine which 5 and I1 of 6. As a preliminary, we discuss an interesting class of matroids introduced in [I whichwill also be useful later see 6.
Definition 6. Any such sequence of pairs parameterizffia connected nested matroid. The class of nested matroids is closed under the following operations Truncation: T N ao, b. Free extension: N a o, Deletion: where pi E N a o, N ao,. Contraction: N ao,.
Duality: If a nested matroidl N of rank n and cardinalFty The hereditary class of nested matroids has,as excluded minors, the sequence Thus, M2 of 4. To get the Tutte polynomial t N a o, In this case, 2. One type e. Thus, n 6. That Wn-i I Vi i s y e t unproved. It is customary to choose the logarithm function, and a unimodal sequence 2 is said to be 10; cmcave if, for all 4.
See [79].
We also note that the analogous result for the log concavity of di was shown by Harper see In particular, show that the log concavity b' i implies the log concavity of Find classes of geometries such as dual paving matroids as we show below which give unimodal or strongly log concave Tutte coefficients or Whitney numbers. We cover our bets by offering the problem of finding a geometry whose Whitney numbers are not unimodal.
We will illustrate some techniques with the next two propositions. We begin with a lemma presenting the combinatorial arguments we will use. Lemma 6. Simplifying the proportions : b;, a s Thus, 5 a'. We a r e now ready t o prove t h e l o g concavity of l o g concavity of if bi. Gi By 6. Symmetrizing t h e Let ad k be fixed. G be a paving matroid. Since, f o r a l l t h e sequence In fact, since am Remarks 6.
The T-G group i n v a r i a n t s below have t h e follow- ing formulas. The number of independent s e t s of rank r: b. The absolute rth Whitney number: c. The reduced rth Whitney number: d. The absolute Miibius function: f. The acyclic or alpha Znvariant and reduced alpha invariant: g.
M1 i s connected. Now, l e t i s connected, be a connected matroid with M1 M1Case 1. If N2 Case 3. Since, i n 3. M i s not the minor i s planar graphic and hence M' i s not, none of t h e above Further, t h e matroids i n 4. M H, El fB This i s proved i n Condition 6.