On the structure of the h-vector of a paving matroid · Example: Uniform matroid E sets sets; r U r;n = (E;I), where E is the set [n] and I= fI E jjIj rg. The bases are the sets of

On the structure of the h-vector of a pavingmatroid

Criel Merino Steve Noble Marcelino Ramırez-IbanezRafael Villarroel

ECCO 2012

Matroids: Definition

A matroid M is an ordered pair M = (E , I) such that E is a finiteset and I is a collection of subsets of E , called independent sets,satisfying

I1) ∅ ∈ I;

I2) if I ∈ I and I ′ ⊆ I , then I ′ ∈ I;

I3) if I1, I2 ∈ I and |I1| < |I2|, ∃e ∈ I2 − I1 s.t. I1 + e ∈ I.

Criel Merino The h-vector of paving matroids

Matroids: Bases and circuits

E

∅

Maximal independents sets are called bases and all have the samecardinality, called the rank of the matroid. Minimal dependents arecalled circuits.



E

∅




E

∅

Maximal independents sets are called bases and all have the samecardinality, called the rank of the matroid.

Minimal dependents arecalled circuits.



E

∅



Example: Uniform matroid

E

Dep

endentsets

Indep

endentsets

∅

r

Ur ,n = (E , I), where E is the set [n] and I = {I ⊆ E | |I | ≤ r}.

The bases are the sets of cardinality r and the circuits the sets ofcardinality r + 1.



E

Dep

endentsets

Indep

endentsets

∅

r

Ur ,n = (E , I), where E is the set [n] and I = {I ⊆ E | |I | ≤ r}.The bases are the sets of cardinality r

and the circuits the sets ofcardinality r + 1.



E

Dep

endentsets

Indep

endentsets

∅

r

r+1

Ur ,n = (E , I), where E is the set [n] and I = {I ⊆ E | |I | ≤ r}.The bases are the sets of cardinality r and the circuits the sets ofcardinality r + 1.


Paving matroid

A matroid M is paving if all it’s circuits have size at least r(M)


Paving matroid

A matroid M is paving if all it’s circuits have size at least r(M)E

Dep

endentsets

Indep

endentsets∅

r(M)


Importance of paving matroids

1973 J. E. Blackburn, H. H. Crapo, and D. A. Higgs. A catalogueof combinatorial geometries.

1976 Dominic Welsh ask if most matroids are paving.

2008 D. Mayhew and G.F. Royle. Matroids with nine elements.

2010 D. Mayhew, M. Newman, D. Welsh, and G. Whittle. Theasymptotic proportion of connected matroids.




















Simplicial complex

Associated to any simplicial complex ∆ we have its f-vector(f0, f1, . . . , fr ) and the generating function

f∆(x) =r∑

i=0

fixr−i .

A simplicial complexes is pure if all its facets have the samecardinality.

Octahedron

f=(1,6,12,8)


Simplicial complex

Associated to any simplicial complex ∆ we have its f-vector(f0, f1, . . . , fr ) and the generating function

f∆(x) =r∑

i=0

fixr−i .

A simplicial complexes is pure if all its facets have the samecardinality.

Octahedron

f=(1,6,12,8)


Simplicial complex

For a pure simplicial complex ∆, a shelling is a linear order of thefacets F1, F2, . . ., Ft such that each facets meets the complexgenerated by its predecessors in a non-void union of maximalproper faces. A complex is said to be shellable if it is pure andadmits a shelling.


Simplicial complex



Simplicial complex



Simplicial complex



Simplicial complex



Simplicial complex



Simplicial complex



Simplicial complex



Simplicial complex



Simplicial complex

Let ∆i denote the subcomplex generated by the facets F1, F2, . . .,Fi . Ri is the unique minimal face of Fi which lies in ∆i −∆i−1.Thus, associated to any shellable simplicial complex ∆ we have itsshelling polynomial

h∆(x) =t∑

i=1

x |Fi\Ri | =r∑

i=0

hixr−i .

and its h-vector (h0, h1, . . . , hr ).


When you evaluate f∆(x − 1) you we get the polynomial h∆(x).Thus, we obtain the following relations among the fi ’s and the hi ’s.

Observation

h0 + h1 + · · ·+ hr = fr .

hk =k∑

i=0

(−1)i+k

(r − i

k − i

)fi .


Example

Octahedron

f=(1,6,12,8)

h = (1, 3, 3, 1)


Matroid complexes

The family of independent sets of a matroid M forms a simplicialcomplex ∆(M) of dimension r(M), called matroid complex. Thefacets of ∆(M) are the bases of M and therefore ∆(M) is pure. A.Bjorner proved that the complex is shellable by using anylexicographic order on the bases.


Example

Octahedron

h∆(x) = x3 + 3x2 + 3x + 1


h-vector for paving matroids

Our question is simple. How is the h-vector of a paving matroid?.

The answer is simple as well.

f -vector is (

(n

0

),

(n

1

), . . . ,

(n

r − 1

), b(M))



Our question is simple. How is the h-vector of a paving matroid?.The answer is simple as well.

f -vector is (

(n

0

),

(n

1

), . . . ,

(n

r − 1

), b(M))



E

Dep

endentsets

Indep

endentsets∅

r(M)

k-subset

Our question is simple. How is the h-vector of a paving matroid?.The answer is simple as well.

f -vector is (

(n

0

),

(n

1

), . . . ,

(n

r − 1

), b(M))



Using the relation between the fi ’s and hi ’s, the h-vector is

(

(n − r − 1

0

),

(n − r

1

), . . . ,

(n − 2

r − 1

), b(M)−

(n − 1

r − 1

)).

Only rest to know

hr = b(M)−(n − 1

r − 1

).



Using the relation between the fi ’s and hi ’s, the h-vector is

(

(n − r − 1

0

),

(n − r

1

), . . . ,

(n − 2

r − 1

), b(M)−

(n − 1

r − 1

)).

Only rest to know

hr = b(M)−(n − 1

r − 1

).



Note that

hk =

(n − r + k − 1

k

).

This expression correspond to the number of monomials over n − rvariables with degree k , for 0 ≤ k ≤ r − 1.


Multicomplex

The set of all monomials over z1, . . . , zd is a poset with thedivisibility relation.

A multicomplex Σ in this poset is a subset ofmonomials which is close under divisibility. If all the maximalelements of Σ have the same degree we said that Σ is pure.


Multicomplex

The set of all monomials over z1, . . . , zd is a poset with thedivisibility relation. A multicomplex Σ in this poset is a subset ofmonomials which is close under divisibility.

If all the maximalelements of Σ have the same degree we said that Σ is pure.


Multicomplex

The set of all monomials over z1, . . . , zd is a poset with thedivisibility relation. A multicomplex Σ in this poset is a subset ofmonomials which is close under divisibility. If all the maximalelements of Σ have the same degree we said that Σ is pure.


Example

x

y

z

x2

xy

y2

yz

z2

xz

x3

x2y

xy2

y3

zy2

yz2

z3

xz2

x2z

x2y2

1

x3z

x2z2 y3z

yz3

xyz xy2z


Multicomplex

Associated to Σ we have its o-vector (o0, o1, . . . , or ), where oi isthe number of monomials of degree i .

A vector (h0, h1, . . . , hr ) is apure O-sequence if there is a pure multicomplex Σ with(h0, h1, . . . , hr ) as o-vector.


Multicomplex

Associated to Σ we have its o-vector (o0, o1, . . . , or ), where oi isthe number of monomials of degree i . A vector (h0, h1, . . . , hr ) is apure O-sequence if there is a pure multicomplex Σ with(h0, h1, . . . , hr ) as o-vector.


Example

x

y

z

x2

xy

y2

yz

z2

xz

x3

x2y

xy2

y3

zy2

yz2

z3

xz2

x2z

x2y2

1

x3z

x2z2 y3z

yz3

xyz xy2z

Its o-vector is (1, 3, 6, 10, 6)


Stanley’s conjecture

In 1977, Richard Stanley made the following conjecture

Conjecture

The h-vector of a matroid complex is a pure O-sequence.



1999 N. Biggs and C. Merino. Cographic Matroids.

2010 J. Schweig. Lattices path matroids.

2010 S. Oh. Cotransversal Matroids.

2010 H. Tai Ha, E. Stokes and F. Zanello. rank-3 matroids.

2011 Y. Kemper, J. De Loera and S. Klee. Corank 2 matroids.






























Stanley’s conjecture and paving matroids

Mr−1,n−r multicomplex with maximal elements

the monomials of degree r − 1 over z1, z2, . . . , zn−rWe define the multicomplex Mr ,d to be the pure multicomplex inwhich the maximal elements are all the monomials of degree r in dindeterminates z1, . . . , zd . This multicomplex gives an O-sequence(o0, . . . , or ), where ok =

(d+k−1k

).



g(r, n) = min {b(M)−(n−1r−1

)|M ∈ P r,n}.


the monomials of degree r − 1 over z1, z2, . . . , zn−r

Now, let us define

g(r , n) = min{hr (M) |M ∈ Pr ,n

}.



f(r, n− r) = min {hr|(h0, . . . , hr) is the O-sequence

g(r, n) = min {b(M)−(n−1r−1

)|M ∈ P r,n}.

ofM⊃M r−1,n−r}.


the monomials of degree r − 1 over z1, z2, . . . , zn−r

Finally, take

f (r , d) = min{or | (o0, . . . or ) is the pure O-sequence ofM⊃Mr−1,d}.


Stanley’s conjecture for paving matroids follows from the followingtheorem.

Theorem

For all rank-r loopless and coloopless paving matroid M with nelements we have that g(r , n) is at least f (r , n − r).


Proof idea

Theorem


We use a careful induction on r + n (and that paving matroids areclose under minors)

Lemma

For r ≥ 1 and d ≥ 1 we have f (1, d) = 1, f (2, d) = dd/2e,f (r , 1) = 1 and f (r , 2) = dr/2e.

Lemma

For r ≥ 2 and d ≥ 2, f (r , d) ≤ f (r , d − 1) + f (r − 1, d).


Proof idea

Theorem



Lemma


Lemma



Proof idea

Theorem



Lemma


Lemma



Proof idea

But, we need a structural result for paving matroids.

Lemma

Let M be a rank-r coloopless paving matroid. If for every elemente of M, M \ e has a coloop, then one of the following three caseshappens.

1 M is isomorphic to Ur ,r+1, r ≥ 1.

2 M is the 2-stretching of a uniform matroid Us,s+2, for somes ≥ 1.

3 M is isomorphic to U1,2 ⊕ U1,2.

Where the 2-stretching of a matroid M is the matroid obtained byreplacing each of the elements of M by 2 elements in series.


Thanks

Documents

On the structure of the h-vector of a paving matroid · Example: Uniform matroid E sets sets; r U r;n = (E;I), where E is the set [n] and I= fI E jjIj rg. The bases are the sets of