Optimal partitions of finite sets: a report on unfinished work in - PowerPoint PPT Presentation

Optimal partitions of finite sets: a report on unfinished work in progress PJC60 at Ambleside: Thursday, 23 August 2007 Peter M Neumann: Queen’s College, Oxford • Introduction: set partitions and Bell numbers • Optimal partitions and the main problem • Inequalities • Some near-theorems and computations • Conclusion 0

Introduction: the Bell numbers Throughout n is a natural number, X a finite set of size n . Recall the Bell number B n := number of set-partitions of X . Recall that B n e e z − 1 = n ! z n . � n � 0 1

Introduction: a useful function Define λ ( n ) by λ e λ = n . Then λ ( n ) = log n − log log n + log log n �� log log n � 2 � + O . log n log n In fact p k (log log n ) � λ ( n ) = log n − log log n + , (log n ) k k � 1 where p k ( t ) is a polynomial of degree k ; leading term t k /k ; alternating signs; obtainable by “boot-strapping”. 2

Introduction: estimates for Bell numbers Many asymptotic estimates for the Bell numbers are known. E.g. B n ∼ n − 1 2 λ ( n ) n + 1 2 e λ ( n ) − n − 1 ; Lov´ asz: log B n = log n − log log n − 1 + log log n De Bruijn: + n log n �� log log n �� log n + 1 1 � log log n � 2 + O . (log n ) 2 2 log n 3

Optimal partitions: notation For a partition µ n write µ = [ m 1 , m 2 , . . . , m k ] to mean: µ has m r parts of size r , so n = � r m r ; • • k is its largest part, so m k � 1 . Then define r ! m r m r ! . � A ( µ ) := So if ρ is a set-partition of X of shape µ then A ( µ ) = | Aut( X ; ρ ) | . n ! � Note. B n = A ( µ ) . µ n 4

Optimal partitions: the main problem Call µ optimal if it minimises A ( µ ). Problem. Which partitions µ of n are optimal? What do they look like asymptotically? How can we compute them for sizable values of n ? 5

Optimal partitions: basic inequalities Proposition. If µ = [ m 1 , m 2 , . . . , m k ] n and µ is optimal then � r + s � ( m r + 1)( m s + 1) if r � = s , m r + s � r � 2 r � m 2 r ( m r + 1)( m r + 2) , � r � r + s � ( m r + s + 1) � if r � = s , m r m s r � 2 r � m 2 r m r ( m r − 1) . � r Note. Call these 3-part inequalities: there are also 4-part inequalities, 5-part inequalities, etc.; sometimes useful. 6

Some near-theorems, I Near-Theorem. Suppose that n is large, µ = [ m 1 , m 2 , . . . , m k ] n , and µ is optimal. Let c := m 1 . Then c e c − 2 � n � ( c + 1) e c +2 , c e � k � ( c + log c ) e . Comment. Too crude! 7

Some near-theorems, II Near-Theorem [K. K¨ orner]. Suppose that n is large, µ = [ m 1 , m 2 , . . . , m k ] n , and µ is optimal. Let c := m 1 . Then m 1 < m 2 < · · · < m c − 1 � m c and m c � m c +1 > m c +2 > · · · > m k − 1 > m k > 0 . That is, µ is unimodal. 8

A method of computation To seek optimal µ for n in the range n 0 � n � n 1 do: find possibilities for m 1 ; then find possibilities for m 2 ; etc.; for each n , for those µ that emerge, find smallest A ( µ ) . Example [ΠMN, hand calculation] for n = 10 , 000 . If m 1 � 6 then n � 9327; if m 1 � 8 then n � 19 , 354; so m 1 = 7 . Then find m 2 = 25 or m 2 = 26 ; If m 2 = 25 then m 3 ∈ { 63 , 64 } , if m 2 = 26 then m 3 ∈ { 62 , 63 , 64 } ; etc. 9

Some computations Computations [K. K¨ orner, using MAPLE on a PC]: • All optimal partitions tabulated for n � 1100. • All optimal partitions tabulated for 10 , 000 � n � 10 , 100; • Method should do 10 5 � n � 10 5 + 100 or even 10 6 � n � 10 6 + 100 in a few hours of computation. 10

Conclusion There’s much more to be done: 11

Conclusion There’s much more to be done: Fully prove the near-theorems. 11

Conclusion There’s much more to be done: Fully prove the near-theorems. Refine and extend the computations. 11

Conclusion There’s much more to be done: Fully prove the near-theorems. Refine and extend the computations. Conjecture. For very large n , if µ = [ m 1 , m 2 , . . . , m k ] n and µ is optimal then m r is close to λ ( n ) r /r ! . What does “close to” mean? Certainly c 1 � m r ÷ λ ( n ) r /r ! � c 2 , perhaps even | m r − λ ( n ) r /r ! | � c 3 . 11

Conclusion There’s much more to be done: Fully prove the near-theorems. Refine and extend the computations. Conjecture. For very large n , if µ = [ m 1 , m 2 , . . . , m k ] n and µ is optimal then m r is close to λ ( n ) r /r ! . What does “close to” mean? Certainly c 1 � m r ÷ λ ( n ) r /r ! � c 2 , perhaps even | m r − λ ( n ) r /r ! | � c 3 . Apply to estimates for Bell numbers B n . 11

Conclusion There’s much more to be done: Fully prove the near-theorems. Refine and extend the computations. Conjecture. For very large n , if µ = [ m 1 , m 2 , . . . , m k ] n and µ is optimal then m r is close to λ ( n ) r /r ! . What does “close to” mean? Certainly c 1 � m r ÷ λ ( n ) r /r ! � c 2 , perhaps even | m r − λ ( n ) r /r ! | � c 3 . Apply to estimates for Bell numbers B n . Happy sixties, Peter 11