Counting Matroids. Rudi Pendavingh. 6 July, Rudi Pendavingh Counting Matroids 6 July, / 23

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1 Counting Matroids Rudi Pendavingh 6 July, 2015 Rudi Pendavingh Counting Matroids 6 July, / 23

2 First steps Definitions [n] := {1,..., n} m n := #{M matroid : E(M) = [n]} m n,r := #{M matroid : E(M) = [n], r(m) = r} Rudi Pendavingh Counting Matroids 6 July, / 23

3 First steps A naive upper bound A matroid on E = [n] is determined by its set of independent sets I 2 E. Hence m n # of subsets of 2 E 2 2n In other words: log m n 2 n (log = logarithm to the base 2) Rudi Pendavingh Counting Matroids 6 July, / 23

4 First steps A better upper bound A matroid on E = [n] of rank r is determined by bases B ( E r ) := {X E : X = r}, so m n,r # of subsets of ( ) E 2 (n r) 2 ( n/2 ) n r Therefore m n = r m n,r (n + 1)2 ( n/2 ) n, hence ( ) ( ) n 1 log m n log(n + 1) + O n 2 n n/2 as n Rudi Pendavingh Counting Matroids 6 July, / 23

5 Bounds of Piff and Knuth Piff s upper bound Lemma The following are equivalent for a matroid M = (E, B) and a set X E X is a dependent set of M there is a circuit C of M so that C X there is a circuit C of M so that X cl M (C) > r M (C) Each matroid M on ground set E is completely determined by K(M) := {(cl M (C), r M (C)) : C a circuit of M} Rudi Pendavingh Counting Matroids 6 July, / 23

6 Bounds of Piff and Knuth Piff s upper bound Each matroid M on ground set E is determined by K(M) := {(cl M (C), r M (C)) : C a circuit of M} Lemma If M is a matroid on n elements, then K(M) 2 n+1 /(n + 1). Proof. The map X cl M (X ) sends C e to cl M (C) for each e C, so Hence K(M) = i<n ( n i #{(cl M (C), r M (C)) : C a circuit of M, r M (C) = i} ( ) n /(i + 1). i ) ( /(i + 1) = n+1 i<n i+1) /(n + 1) = 2 n+1 /(n + 1). Rudi Pendavingh Counting Matroids 6 July, / 23

7 Bounds of Piff and Knuth Piff s upper bound Theorem (Piff, 1973) ( ) log(n) log m n O 2 n as n n Proof. m n is at most # of K 2 E [n + 1] of size K 2 n+1 /(n + 1), so m n 2 n+1 /(n+1) j=0 ( 2 n ) (n + 1) i The theorem follows using k j=0 ( ) N j ( ) en k k Rudi Pendavingh Counting Matroids 6 July, / 23

8 Bounds of Piff and Knuth How to (not) find a lower bound Piff: log m n O(2 n log(n)/n) as n class upper bound log u.b. ( graphic matroids n+1 ) n 2 O(n log n) binary matroids (2 n ) n n 2 GF(q)-representable matroids (q n ) n n 2 log q transversal matroids (2 n ) n n 2 real-representable matroids (Alon, 1986) 2 n3 n 3 Not even close. Rudi Pendavingh Counting Matroids 6 July, / 23

9 Bounds of Piff and Knuth Paving matroids Definition A matroid M is paving if C r(m) for each circuit C of M. Crapo and Rota (1970) consider it likely that paving matroids... would actually predominate in any asymptotic enumeration of geometries based on the enumeration of matroids up to 8 elements (Blackburn, Crapo, and Higgs). Definition A matroid M is sparse paving if both M and M are paving. If almost all matroids are paving, then almost all matroids are sparse paving. Rudi Pendavingh Counting Matroids 6 July, / 23

10 Bounds of Piff and Knuth The Johnson graph J(E, r) is the undirected graph with vertices: ( ) E := {X E : X = r} r and edges: X Y : X Y = 2 We write J(n, r) := J([n], r). J(4, 2) Lemma S is a stable set of J(E, r) M = (E, ( ) E \ S) is a sparse paving matroid r Rudi Pendavingh Counting Matroids 6 July, / 23

11 Bounds of Piff and Knuth Definitions s n := #{M sparse paving matroid : E(M) = [n]} s n,r := #{M sparse paving matroid : E(M) = [n], r(m) = r} s n,r equals the number of stable sets of J(n, r) Rudi Pendavingh Counting Matroids 6 July, / 23

12 Bounds of Piff and Knuth Knuth s lower bound on the number of matroids Let α(g) := max{ S : S V (G) a stable set of G}. Theorem (Knuth, 1974) log s n,r α(j(n, r)) Proof. J(n, r) has a stable set S 0 of size α(j(n, r)). Each S S 0 is a stable set of J(n, r), so that s n,r 2 S 0 = 2 α(j(n,r)) In 1974, this gave the lower bound log m n log s n log s n,n/2 α(j(n, n/2)) 1 ( ) n 2n n/2 Rudi Pendavingh Counting Matroids 6 July, / 23

13 Bounds of Piff and Knuth An improvement of Knuth s bound Theorem (Graham & Sloane, 1980) J(n, r) has a stable set of size ( n r) /n. Proof. For each k Z, the set ( ) [n] S(n, r, k) := {X : x = k mod n} r x X is stable in J(n, r). Since n k=1 S(n, r, k) = ( ) ( [n] r, we have S(n, r, k) > n ) r /n for some k. And so we obtain log m n log s n,n/2 α(j(n, n/2)) 1 n ( ) n. n/2 Rudi Pendavingh Counting Matroids 6 July, / 23

14 Recent work Three recent bounds By counting certain compressed, faithful descriptions of matroids: Theorem (Bansal, P., van der Pol, 2014) ( log 2 ( )) (n) n log m n O n n/2 Using Shearer s Entropy Lemma: Theorem (Bansal, P., van der Pol, 2014) ( ( )) log(n) n log m n O n n/2 Adapting a counting method for stable sets in graphs: Theorem (Bansal, P., van der Pol, 2014) log m n 2 + o(1) n as n as n ( ) n as n n/2 Rudi Pendavingh Counting Matroids 6 July, / 23

15 Recent work Compressed matroid descriptions Bounding by compression By Piff, Knuth, Graham&Sloane: ( ) 1 n log s n log m n O( log(n) 2 n ) as n n n/2 n Theorem (Bansal, P., van der Pol, 2014) ( log 2 ( )) (n) n log m n O n n/2 as n Proof outline: a cover of a matroid M gives a compressed description of M each matroid on n elements has a cover of size k n O ( log(n) n ( n ) ) n/2 # of matroids on n elements of rank r is the # of covers of size k n Rudi Pendavingh Counting Matroids 6 July, / 23

16 Recent work Compressed matroid descriptions Matroid covering A flat F such that F X > r(f ) certifies that X is dependent. Definition A set of flats Z covers a set U 2 E if F X > r(f ) for each dependent X U If Z covers ( E r ), then {(F, rm (F )) : F Z} is a faithful description of M of rank r. Lemma Each matroid M on n elements has a cover Z such that Z 10 log(n) n Proof. ( n ) n/2 J(n, r) is d = r(n r)-regular, so has a dominating set D of size D ln(d+1)+1 d+1 For each X D there is a set Z X covering N(X ) {X } of size Z X min{r, n r}. Then Z := X D Z X covers ( E r ), and Z min{r, n r} ln(d+1)+1 d+1 ( n r ). ( n ) r. Rudi Pendavingh Counting Matroids 6 July, / 23

17 Recent work Entropy Entropy counting Let W 2 U. For T U, put W T := {W T : W W}. If T 1,..., T m partitions U, then: W W T1 W Tm The following is a consequence of Shearers Entropy Lemma: Theorem If T 1,..., T m U and each u U occurs in at least k of the sets T i, then Lemma (Bansal, P., van der Pol) W k W T1 W Tm If m n,r counts the number of matroids in a contraction-closed class, then log(1 + m n,r ) ( n r ) log(1 + m n t,r t) ) U := ( ) E r, W := {sets B satisfying base exchange}, TS := {X ( ) ( E r : S X } for S E ) t. ( n t r t Rudi Pendavingh Counting Matroids 6 July, / 23

18 Recent work Entropy Bounding by entropy Lemma (Bansal, P., van der Pol) If m n,r counts the number of matroids in a contraction-closed class, then log(1 + m n,r ) ( n r Theorem (Bansal, P., van der Pol, 2014) ) log(1 + m n t,r t) ) log m n O( log(n) n ( n t r t ( ) n ) as n n/2 Proof. log(1 + m n,r ) ( n r ) log(1 + m n r+2,2) ) ( n r log(n + 1) n Rudi Pendavingh Counting Matroids 6 July, / 23

19 Recent work Conjectures Conjectures on matroid asymptotics We say that asymptotically almost all matroids have property P if #{M M n : M has property P} lim = 1 n #M n Here M n denotes the set of matroids with ground set {1,..., n}. Conjecture (Mayhew, Newman, Welsh and Whittle, 2011) If N is a fixed sparse paving matroid, then a.a.a. matroids have N as a minor. Let k N. Asymptotically almost all matroids are k-connected. Asymptotically almost all matroids are sparse paving. Asymptotically almost all matroids M on n elements have n 2 r(m) n 2. Rudi Pendavingh Counting Matroids 6 July, / 23

20 Recent work Conjectures Size of minor-closed classes Conjecture (Mayhew, Newman, Welsh and Whittle, 2011) If N is a fixed sparse paving matroid, then a.a.a. matroids have N as a minor. Theorem (P., van der Pol, 2013) U 2,k U 3,6 P 6 Q 6 R 6 True if N = U 2,k for some k 2, or if N is one of U 3,6, P 6, Q 6 or R 6. Theorem (P., van der Pol, 2014) The number s n of sparse paving matroids on n elements without M(K 4 )-minor satisfies log s n 1 ( ) n (1 o(1)) as n n n/2 Rudi Pendavingh Counting Matroids 6 July, / 23

21 Recent work Conjectures Connectivity Conjecture (Mayhew, Newman, Welsh and Whittle, 2011) Let k N. Asymptotically almost all matroids are k-connected. Theorem (Mayhew, Newman, Welsh and Whittle, 2011) Asymptotically almost all matroids are connected. Theorem (Oxley, Semple, Warshauer, Welsh, 2013) Asymptotically almost all matroids are 3-connected. Using Entropy: Theorem (P., van der Pol, 2014) Let k N. There is a c such that a.a.a matroids on n elements have no (< k)-separation {A, B} with min{ A, B } c log 2 (n). Rudi Pendavingh Counting Matroids 6 July, / 23

22 Recent work Conjectures Sparse paving matroid vs. matroid Conjecture (Mayhew, Newman, Welsh and Whittle, 2011) Asymptotically almost all matroids are sparse paving. So far: ( ) 1 n log s n log m n O n n/2 ( log n n ( )) n n/2 as n Theorem (Bansal, P., van der Pol, 2014) Theorem (P., van der Pol, 2015) log m n 2 + o(1) n ( ) n as n n/2 log m n = (1 + o(1)) log s n as n Rudi Pendavingh Counting Matroids 6 July, / 23

23 Recent work Conjectures The rank of a typical matroid Conjecture (Mayhew, Newman, Welsh and Whittle, 2011) Asymptotically almost all matroids M on n elements have n 2 r(m) n 2 Theorem (Oxley, Semple, Warshauer, and Welsh, 2013) For all ɛ > 0, asymptotically almost all matroids M on n elements have Theorem (P., van der Pol, 2015) 1 2 n ɛn < r(m) < 1 2 n + ɛn If β > , then asymptotically all matroids M on n elements have 1 2 n β n r(m) 1 2 n + β n Rudi Pendavingh Counting Matroids 6 July, / 23