Integer Partition Poset

Transcription

1 Integer Partition Poset Teena Carroll St. Norbert College Presented at the Clemson REU Summer 2010

2 Integer Partitions Given an integer, how can we break it down into the sum of positive integers? ex: n=5 5= = = = =1+4 5=2+3 5=5

3 Ground Rules: Order of summands doesn t matter: 5=2+3 is considered the same partition as 5=3+2. For convenience, we always list the elements of the partition in increasing order. Notation: (2,3) is a partition of n=5

4 Comparing Partitions For a fixed n, let P 1 and P2 be two partitions We say that P 1 covers P2 if we can take one summand in and split it into exactly two parts (then reorder them into increasing order). Notation: P 2 P 1 For n=8, let P 1 =(1,2,5) and P2=(1,1,2,4) then P1 covers P2

5 Organizing Information For a fixed n, we can draw the Hasse Diagram by listing all of the partitions and drawing a line between P1 and P2 whenever P1 covers P2 (5) P 5 (1,4) (2,3) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)

6 Some things to note (5) (1,4) (2,3) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1) Partitions are arranged in levels so all partitions on the same level have the same number of summands. We call the set of all partitions with the same number of parts a level set in the Hasse diagram

7 Vocabulary Pn is the set of all partitions of the integer n with the ordering induced by the transitive closure of our covering relation. We call this relation Pn is a partially ordered set (poset) under (therefore the operation is reflexive, transitive and antisymmetric)

8 What does that mean?! (5) (1,4) (2,3) (1,2,2) (1,1,3) You get there from here? P 1 P 2 if there is a path using only upward edges between P 1 to P 2 (1,1,1,2) We see that (1,1,1,2) (1,1,1,1,1) and (2,3) are related

9 What does that mean?! (5) (1,4) (2,3) (1,2,2) (1,1,3) You can t get there from here. P 1 P 2 if there is a path using only upward edges between P 1 to P 2 (1,1,1,2) We see that (1,2,2) (1,1,1,1,1) and (1,1,3) are not related. We say they are incomparable.

10 (7) (16) (25) (34) (115) (124) (133) (223) (1114) (1123) (1222) (11113) (11122) A selection of elements in the poset which are (111112) all related to each other is called a chain. ( )

11 (7) (16) (25) (34) (115) (124) (133) (223) (1114) (1123) (1222) (11113) (11122) A chain where no elements can be added (111112) to make it longer is called a maximal chain. ( )

12 Antichains Definition: In a partially ordered set, a collection of pairwise incomparable elements is called an antichain. Recall: A level set is a set of elements who all have the same rank. Fact: Every level set forms an antichain.

13 (7) (16) (25) (34) Two examples of antichains. (115) (124) (133) (223) One is a level set (1114) (1123) (1222) one is not a level set. (11113) (11122) (111112) ( )

14 (7) (16) (25) (34) Two examples of antichains. (115) (124) (133) (223) One is a level set (1114) (1123) (1222) one is not a level set. (11113) (11122) Both are maximal nothing can be added to (111112) either to make it a larger antichain ( )

15 THE QUESTION Definition: A ranked poset is Sperner if the size of the largest antichain is the size of the largest level set. Open question: Is the integer partition poset Sperner for all n?

16 History and Background Canfield (2003) Computationally verified that Pn is Sperner for all n<46.

17 (7) (16) (25) (34) (115) (124) (133) (223) (1114) (1123) (1222) A chain partition (11113) (11122) is a selection of chains so (111112) every element of the poset appears in exactly one chain ( )

18 Big Idea Dilworth s Theorem (1950) For any poset, the maximum size of an antichain is the minimum size of a chain partition.

19 (7) (16) (25) (34) (115) (124) (133) (223) (1114) (1123) (1222) Revisiting this picture... (11113) (11122) (111112) ( )

20 (7) (16) (25) (34) (115) (124) (133) (223) In light of Dilworth s Theorem (1114) (1123) (1222) (11113) (11122) (111112) ( ) This is a visual proof that P 7 is Sperner. Since the largest level set has size 4 and I have provided a chain partition of size 4.

21 Big Idea Revisited Sperner restated in terms of Dilworth s Theorem For a ranked poset, a poset is Sperner if there is a chain partition the same size as the largest level set.

22 So exactly how big is this largest level set, Teena? Well, that isn t really important right now all you need to know is that we understand this quantity well...

23 History and Background Definition: A ranked poset is unimodal if there is a level set of maximum size and the size of the level sets increases until it reaches that level and decreases afterwards. * neither increasing or decreasing is strict here

24 Unimodality Szekeres showed that for large enough n, P n is unimodal. It has been computationally verified that for n<2000 Pn is unimodal. No combinatorial proof of unimodality of P n is known.

25 Bipartite Matching Property Definition: A ranked poset has the bipartite matching property if given any two consecutive levels, there is a matching using the edges of the poset from the smaller level into the larger level

26 The largest level set Theorem: If a ranked poset has the bipartite matching property and it is unimodal, it is Sperner.

27 Canfield (2003) History and Background Computationally verified that Pn is Sperner for all n<46 by showing they have the bipartite matching property. i.e. For every pair of consecutive levels, they verified that there is a matching using the edges of the poset

28 How did they verify these matchings exist? A C program considered each pair of levels and answered yes or no. They listed all elements lexicographically (i.e. alphabetically for strings of numbers--- 1 comes before 2 etc.) Picked an initial matching and used an alternating path algorithm to find a matching which saturated the smaller side.

29 A step in the alternating path process

30 First Question Canfield (2003) Computationally verified that Pn is Sperner for all n<46. What happened at 46?

31 Questions What do the chain partitions look like when you put the matchings together? Are there any similarities between the matchings chosen for different levels for a fixed n? Are there any similarities between values of n in terms of how the chain partitions are formed? How sensitive is the outcome to the initial matching we choose?

32 How do we start looking for Patterns? Do the chain partitions treat special elements in a consistent way? elements with parts repeated many times have a low number of edges going up. elements with lots of 1 s have low number of edges going down.

33 How do we start looking for Patterns? How do the chains begin and end? Since every chain has a bottommost element, if we can describe this set we can count it. Since every chain has a topmost element, if we can describe this set we can count it.

34 How do we start looking for Patterns? If we look at many (all?) optimal chain partitions for a given n, Maybe one of them has a clear generalizable description of the top/bottom sets. Maybe we can find a core of elements which are in most (all?) of the top/bottom sets. How many chain partitions are there anyway?

35 How do we start looking for Patterns? If we look a fixed partition for a given n, we can ask how many chains of each length are there? If we look over many (all?) optimal chain partitions we can ask how much variability is there in the chain length profile?

36 There is a lot of data... Does it call for a database? If you have a database which stores chain partitions, you can query a database with new questions as they come up without writing new code.

37 Making the Problem Smaller Can we fix a certain chain and find a chain partition of the rest of the poset? Can we use edges only of a certain type? Can we find a method to chain partition the bottom of the poset, so we can computationally only worry about the top?

38 (7) (16) (25) (34) (115) (124) (133) (223) (1114) (1123) (1222) (11113) (11122) (111112) ( )