Some counting problems related to A relational structure M consists - PDF document

☎ ☎ ☎ ✄ Three counting problems: 1 Some counting problems related to A relational structure M consists of a set X and a family of relations on X . permutation groups The age of M is the class of finite relational Peter J. Cameron structures (in the same language) embeddable in M . School of Mathematical Sciences Problem. How many (a) labelled , (b) unlabelled Queen Mary and Westfield College � M structures in Age ✁ ? London E1 4NS, U.K. p.j.cameron@qmw.ac.uk ✂ 1 ✄ 2 ✄ n [Labelled structures have the element set ✆ . Unlabelled structures are isomorphism types.] 1 3 Three counting problems: 2 A permutation group G on a set X is oligomorphic if ‘I count a lot of things that there’s no need to count,’ G has only finitely many orbits on X n , for all n : Cameron said. ‘Just because that’s the way I am. But equivalently, on the set of n -subsets of X , or on the I count all the things that need to be counted.’ set of n -tuples of distinct elements of X . Richard Brautigan, The Hawkline Monster Problem. How many orbits on (a) n -sets, (b) n -tuples of distinct elements, (c) all n -tuples? 2 4

✄ ☎ ☎ ☎ ✞ ✞ Connections: 12 Three counting problems: 3 The structure M is homogeneous if any isomorphism Let T be a complete consistent theory in the between finite induced substructures of M . first-order language L . An n - type over T is a set S of formulae in L with free variables x 1 ✄ x n , maximal Fra¨ ıss´ e’s Theorem : A class of finite structures is subject to being consistent with T . the age of a countable homogeneous structure M if and only if it is closed under isomorphism, closed We say that T is ℵ 0 - categorical if it has a unique under taking induced substructures, contains only countable model (up to isomorphism). This is countably many members up to isomorphism, and equivalent to there being only finitely many n -types has the amalgamation property . for each n (the theorem of Engeler, Ryll-Nardzewski and Svenonius). If these conditions hold, then M is unique up to e class and M its isomorphism. We call a Fra¨ ıss´ Problem. How many n -types? Fra¨ ıss´ e limit . 5 7 Connections: 12 An example There is a natural topology on the symmetric group of countable degree (pointwise convergence) with the Let M be the unique countable dense totally ordered properties that ✝ . set (a) a subgroup is closed if and only if it is the By Cantor’s Theorem , its theory is ℵ 0 -categorical. automorphism group of a homogeneous relational structure; Its age consists of all finite ordered sets: there is one unlabelled structure, and n ! labelled structures, on n (b) the closure of a subgroup is the largest overgroup with the same orbits on X n for all n . elements. Its automorphism group is transitive on n -sets for Hence counting labelled/unlabelled structures in a every n . Fra¨ ıss´ e class is equivalent to counting orbits of a permutation group on n -sets/ n -tuples of distinct elements. 6 8

✑ ✎ ✟ ✠ ✟ ✟ ✁ ✟ ✑ ✍ ✠ ✠ ✌ ✁ ✟ ☛ ☛ ✟ ✁ ✟ ✁ ✟ ✛ ✗ ✟ ✟ ✁ ✙ ✙ ✟ ✗ ✟ ✁ ✔ ✕ ✔ ✔ ✓ ✠ Three counting sequences Connections: 23 Which sequences occur? Let ☞ and be the sets of The theorem of Engeler, Ryll-Nardzewski and f - and F -sequences for oligomorphic groups. A Svenonius says more than we have seen so far: compactness argument shows that both are closed in in the topology of pointwise convergence, so the (a) for a countable structure M , the theory of M is conditions should be local ones! � M ℵ 0 -categorical if and only if Aut ✁ is oligomorphic; Theorem: f n f n for all n . (Similarly F n F n but ✏ 1 ✏ 1 (b) if these condition holds, then all n -types are this is trivial.) realised in M , and two n -tuples realise the same type � M if and only if they are in the same orbit of Aut ✁ . 1 , F n n ! , and Example: total orders. f n n � n ∑ Thus, if T is ℵ 0 -categorical, counting n -types of T is F S ✄ k ✁ k ! n � T ✡ 1 equivalent to counting orbits of Aut k ✁ on n -tuples. is the number of labelled preorders on n points. 9 11 Three counting sequences Growth rates: examples ✒ n Let G be an oligomorhic permutation group on X . Let � S k 1 k Polynomial : for example, f n is a 1 k � G f n no. of G -orbits on n -subsets; 1 in n . polynomial of degree k � G � S Wr S � n F n no. of G -orbits on n -tuples of distinct Fractional exponential : e.g. f n p ✁ , the � n 1 ✖ 2 partition function (roughly exp elements; ✁ ). � G ✘ 5 ✖ 2 c n , F no. of G -orbits on n -tuples. Exponential : e.g. for boron trees, f n an n where c 2 ☎ 483 ✙ . Then f n and F n count unlabelled and labelled � S 2 Wr A n -element structures in a Fra¨ e class, while F ıss´ Another example: f n F n , the n th Fibonacci n counts n -types in an ℵ 0 -categorical theory. We have number. n � n � n n ! . ∑ Factorial : e.g. two independent total orders, f n F S ✄ k ✁ F k , where S ✄ k ✁ is the Stirling number n ✡ 1 k of the second kind; Exponential of polynomial : e.g. graphs, ✜ n ! . 2 n ✚ n ✘ 1 ✖ 2 f n n ! f n . f n F n 10 12

✦ ✤ ✢ ✢ ✢ ✢ ✢ ✢ ✢ ✢ ✣ ✣ ✣ ✣ ✣ ✣ ✤ ✤ ✟ ✟ ✦ ✥ ✦ ☛ ☛ ✟ ✤ ✔ ✁ ☎ ✤ ✤ ✤ ✤ ✤ ✦ ✢ ✣ ✣ ✢ ✢ ✢ ✢ ✢ ✤ ✢ ✟ ✟ ✁ ✟ ✁ ✣ ✤ ✤ ✤ ✤ ✤ ✣ ✣ ✣ ✣ ✣ ✣ ✢ ✢ ✢ ✢ ✢ ✢ ✦ Smoothness Boron trees Sequences arising from groups should grow smoothly. In particular, for polynomial growth, ✜ log n should tend to a limit; for fractional log f n A boron tree is a tree in which all vertices have ✜ log n for fractional exponential, valency 1 or 3 . The leaves (‘hydrogen atoms’) of a exponential, loglog f n ✜ n for exponential, etc. How do you state a log f n boron tree carry a quaternary relation. The class of such relational structures is a Fra¨ ıss´ e class. general conjecture? A specific question. Define an operator S on sequences by Sa b if ∞ ∞ � 1 ∑ ∏ b n x n x k ✘ a k n ✡ 0 ✡ 1 k ✜ f n tends Is it true that, if f Sa counts orbits, then a n to a limit (possibly 0 or 1 )? 13 15 Growth rates: restrictions Smoothness Pouzet: For homogeneous binary relational structures, either � f n � G � f n � G Wr S Remark 1. If f ✁ then S f ✁ . ✍ , c 1 c 1 n d c 2 n d (for some d f n ✄ c 2 0 ), or Similar sequence operators can be defined with any oligomorphic group replacing S . The same conjecture f n grows faster than polynomially. could be made for any such operator. Similarly one could replace wreath products by direct products. � n 1 ✖ 2 ✘ ε exp Macpherson: In the latter case, f n ✁ for � ε n n 0 Remark 2. The operator S has various interpretations ✁ . (see later). 1 for all Macpherson: If G is primitive, then either f n c n for all sufficiently large n , where c 1 . n , or f n 14 16

✟ ☎ ✓ ✟ ✁ ✬ ✯ ✟ ★ ✁ ✄ ★ ✟ ✯ ✩ ✟ ★ ★ ✔ ✓ ★ ✑ ✁ An algebra Let X be an infinite set. For any non-negative integer Integral domain? n , let V n be the set of all functions from the set of n -subsets of X to ✧ . This is a vector space over ✧ . G is I conjecture that, if G has no finite orbits, then an integral domain. Define V n This would have as a consequence a smoothness � f n n ✪ 0 result for the sequence ✁ , in view of the following with multiplication defined as follows: for f ✥ V m , result, in view of the following: g ✥ V n , let fg be the function in V m ✏ n whose value on � m n ✁ -set A is given by the V n be a graded algebra which is an Let � V n � A � B � A integral domain, with dim a n . Then ∑ fg f ✁ g ✮ B 1 for all m a m a m a n ✄ n . B ✫ A ✏ n ✬ B ✭ m This is the reduced incidence algebra of the poset of finite subsets of X . 17 19 Polynomial algebra? � M Aut Let M be the Fra¨ ✞ , and G ıss´ e limit of ✁ . Under the following hypotheses, it can be shown that G is a polynomial algebra: An algebra ✰ there is a notion of disjoint union in ✞ ; G be the If G is a permutation group on X , let ✪ 0 V G n , where V G subalgebra of A of the form ✰ there is a notion of involvement on the n -element n is n the set of functions fixed by G . structures in ✞ , so that if a structure is partitioned, it involves the disjoint union of the induced � V G If G is oligomorphic, then dim ✁ is equal to the substructures on its parts; n � G number F n ✁ of orbits of G on n -sets. ✰ there is a notion of connected structure in ✞ , so that every structure is uniquely expressible as the disjoint union of connected structures. � M The polynomial generators of A ✁ are the characteristic functions of the connected structures. 18 20