Ying Gao. May 18th, rd Annual PRIMES Conference. Depths of Posets Ordered by Refinement. Ying Gao. Mentored by Sergei Bernstein.

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1 of of 3rd Annual PRIMES Conference May 18th, 2013

2 of Partially-ordered sets, or posets, are sets in which any two elements may be related by a binary relation.

3 of Partially-ordered sets, or posets, are sets in which any two elements may be related by a binary relation. For elements A, B, C of a poset, 1. A A (reflexivity); 2. If A B and B A, then A = B (anti-symmetry); 3. If A B and B C, then A C (transitivity).

4 of Partially-ordered sets, or posets, are sets in which any two elements may be related by a binary relation. For elements A, B, C of a poset, 1. A A (reflexivity); 2. If A B and B A, then A = B (anti-symmetry); 3. If A B and B C, then A C (transitivity). may be represented by Hasse diagrams, in which elements A and B are connected, with A below B, if A < B and there is no element C such that A < C < B.

5 of Examples The set of the first 6 natural numbers, ordered by divisibility.

6 of Examples The set of the first 6 natural numbers, ordered by divisibility. Hasse Diagrams

7 of Examples The set of subsets of {1, 2}, ordered by inclusion.

8 of Examples The set of subsets of {1, 2}, ordered by inclusion. Hasse Diagrams (1, 2) (1) (2) ( )

9 of Intervals An interval I = [A, B] of a poset includes all elements C such that A C B.

10 of Intervals An interval I = [A, B] of a poset includes all elements C such that A C B. Example The color blue represents the interval. F H C D E G A B An interval I = [A, G]

11 of Interval partitions of a poset In an interval partition, the poset is completely partitioned into non-overlapping intervals. Each element of the poset is in exactly one interval. Example F H G C D E A B An interval partition P

12 of Depth Definition The depth of an element X 0 in a poset is defined to be the maximum possible number of elements in a chain X 0 > X 1 > X 2 > > X n from X 0 to the bottom of the poset.

13 of Depth Definition The depth of an element X 0 in a poset is defined to be the maximum possible number of elements in a chain X 0 > X 1 > X 2 > > X n from X 0 to the bottom of the poset. We say that a level n contains all elements of depth n. Example F H G Level 4 Level 3 C D E Level 2 A B Level 1

14 of ndepth For an interval I, [I ] = max(depth[x ]) over all elements X in I. For I = [A, B], [I ] = depth[b]. F H C D E G A B Interval I = [A, G]. [I ] = depth[g] = 3.

15 of ndepth For a partition P of a poset, [P] = min([i ]) over all intervals I in P. Which interval in the partition has the least depth? F H C D E G A B [P] = [B, E] = 2.

16 of ndepth For a poset G, [G] = max([p]) over all possible partitions P of G. Which partition(s) of the poset have the greatest depth? H H F G F G C D E C D E A B A B = 2 = 3 [G] = 3.

17 of Past work Yinghui Wang: of a product of chains n k \ 0 is (n 1) k/2 Biro, Howard, Keller, Trotter, and Young For a poset B of the non-empty subsets of an n-element set ordered by inclusion, [B] n/2 We will study the properties of posets comprised of partitions of sets ordered by.

18 of Set-Partitions A partition of a set is the division of the set of distinct points into subsets. Every element of the set is partitioned into some subset, and no element is within two or more subsets. Examples Partitions of a 6-element set.

19 of It is possible to order partitions of a set by. A partition of a set P b is considered finer than another partition P a if all subsets within P b are within some subset in P a. If P b is finer than P a, P a is coarser than P b. Examples Right partition is finer than left partition

20 of of the ordering These posets depend solely on the size of the set that is partitioned Let G i denote a poset ordered by of all set partitions of the set with i elements except the empty partition, the partition of each element into a separate subset. Example G 3.

21 of Now that we know what a poset ordered by looks like... How do we find the of such a poset? First, we need a few tools.

22 of A rotation of a certain partition is any other partition that may be obtained by rotating the partition around a circle. A rotation of an interval is the interval formed by rotating each element of the interval the same number of times.

23 of A rotation of a certain partition is any other partition that may be obtained by rotating the partition around a circle. A rotation of an interval is the interval formed by rotating each element of the interval the same number of times. Examples

24 of of set partitions It is possible to group set partitions into classes based on the sizes of subsets in the set partitions. The class C = (S 1, S 2, S 3,..., S n ) with all S i positive integers and S 1 S 2 S 3... S n includes all partitions which consist of exactly n subsets of size S 1, S 2,..., S n. All rotations of a partition are in the same class as the partition.

25 of of set partitions It is possible to group set partitions into classes based on the sizes of subsets in the set partitions. The class C = (S 1, S 2, S 3,..., S n ) with all S i positive integers and S 1 S 2 S 3... S n includes all partitions which consist of exactly n subsets of size S 1, S 2,..., S n. All rotations of a partition are in the same class as the partition. Examples

26 of [G 5 ] A B C (A) x1 [5] (B) x5 [4,1] (C) x10 [3,2] D F E (D) x15 [2,2,1] (E) x10 [3,1,1] (F) x10 [2,1,1,1] Poset of classes in G 5

27 of [G 5 ] In order for it to be true that [G 5 ] = L... It must be impossible to partition the poset so that all intervals I have [I ] > L It must be possible make a partiton P of G 5 such that each interval I in P has [I ] L

28 of [G 5 ] In order for it to be true that [G 5 ] = L... It must be impossible to partition the poset so that all intervals I have [I ] > L It must be possible make a partiton P of G 5 such that each interval I in P has [I ] L Lemma [G 5 ] < 4.

29 of [G 5 ] A B C (A) x1 [5] (B) x5 [4,1] (C) x10 [3,2] D F E (D) x15 [2,2,1] (E) x10 [3,1,1] (F) x10 [2,1,1,1] Poset of classes in G 5

30 of [G 5 ] We attempt to show that [G 5 ] = 3 by constructing a partition P of G 5 with 3. It is possible to make a partition P with ten non-overlapping intervals of the form [F, C] and five non-overlapping intervals of the form [D, B], as well as an interval [A, A], which will completely partition the poset into intervals of 3 or greater. Theorem The of G 5 is 3.

31 of [G 5 ] C C B D E D E D F The 10 intervals of the form [F, C] and 5 intervals of the form [D, B] include all the rotations of these three intervals. F

32 of The values of [G i ] i [G i ]

33 of The values of [G i ] i [G i ] [G i ] is non-decreasing; for all i, [G i+1 ] [G i ]

34 of The values of [G i ] i [G i ] [G i ] is non-decreasing; for all i, [G i+1 ] [G i ] [G i ] does not increase very fast; for all i, [G i+1 ] [G i ] + 1

35 of The values of [G i ] i [G i ] [G i ] is non-decreasing; for all i, [G i+1 ] [G i ] [G i ] does not increase very fast; for all i, [G i+1 ] [G i ] + 1 [G i ] increases infinitely; for any L, there is some i large enough that [G i ] = L.

36 of Future Work Describe upper and lower bounds for [G i ] for very large i Prove that [G i ] increases linearly Concretely describe the sequence [G i ]

37 of Thanks to My mentor,, for his patience and guidance Professor Richard Stanley, for suggesting the project Tanya Khovanova and Dai Yang, for their advice PRIMES, for providing this research opportunity My parents, for their continuing support

38 of Bibliography Wang, Yinghui. (2010). A New Depth Related to the Stanley Depth of Some Power Sets of Multisets. Online. Biro, C. Howard, D. Keller, M. Trotter, W. Young, S. Interval Partitions and Stanley Depth. Online. Stanley, Richard. (2011). Enumerative Combinatorics, Volume 1, Second Edition,