Depths of Posets Ordered by Refinement

Depths of Posets Ordered by Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths Depths of Posets Ordered by Reﬁnement Past Work

Refinement Tools for finding Ying Gao ndepth Rotations Mentored by Sergei Bernstein Classes Ndepth of a refinement 3rd Annual PRIMES Conference -ordered poset Results May 18th, 2013 Conclusion

Acknowledgments • 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). • Posets 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.

Depths of Posets Ordered by Posets Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths • Partially-ordered sets, or posets, are sets in which any Past Work two elements may be related by a binary relation ≤ . Reﬁnement

Tools for ﬁnding ndepth Rotations Classes Ndepth of a reﬁnement -ordered poset

Results

Conclusion

Acknowledgments • Posets 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.

Depths of Posets Ordered by Posets Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Posets Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths • Partially-ordered sets, or posets, are sets in which any Past Work two elements may be related by a binary relation ≤ . Refinement • For elements A, B, C of a poset, Tools for finding ndepth 1. A ≤ A (reflexivity); Rotations Classes 2. If A ≤ B and B ≤ A, then A = B (anti-symmetry); Ndepth of a 3. If A ≤ B and B ≤ C, then A ≤ C (transitivity). refinement -ordered poset • Posets may be represented by Hasse diagrams , in which Results elements A and B are connected, with A below B, if Conclusion A < B and there is no element C such that A < C < B. Acknowledgments Hasse Diagrams 4 6

2 3 5

Depths of Posets Ordered by Examples Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets • The set of the ﬁrst 6 natural numbers, ordered by Depths Past Work divisibility.

Reﬁnement

Tools for ﬁnding ndepth Rotations Classes Ndepth of a reﬁnement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Examples Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets • The set of the ﬁrst 6 natural numbers, ordered by Depths Past Work divisibility.

Reﬁnement

Tools for ﬁnding Hasse Diagrams ndepth Rotations 4 6 Classes Ndepth of a reﬁnement -ordered poset Results 2 3 5 Conclusion

Acknowledgments

1 Hasse Diagrams

(1, 2)

(1) (2)

(∅)

Depths of Posets Ordered by Examples Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction • The set of subsets of {1, 2}, ordered by inclusion. Posets Depths Past Work

Reﬁnement

Tools for ﬁnding ndepth Rotations Classes Ndepth of a reﬁnement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Examples Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction • The set of subsets of {1, 2}, ordered by inclusion. Posets Depths Past Work Hasse Diagrams Reﬁnement

Tools for ﬁnding ndepth (1, 2) Rotations Classes Ndepth of a reﬁnement -ordered poset Results (1) (2) Conclusion

Acknowledgments

(∅) Example The color blue represents the interval.

F G

C D E

A B

An interval I = [A, G]

Depths of Posets Ordered by Intervals Reﬁnement

Ying Gao Mentored by Sergei Bernstein • An interval I = [A, B] of a poset includes all elements C

Introduction such that A ≤ C ≤ B. Posets Depths Past Work

Reﬁnement

Tools for ﬁnding ndepth Rotations Classes Ndepth of a reﬁnement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Intervals Reﬁnement

Ying Gao Mentored by Sergei Bernstein • An interval I = [A, B] of a poset includes all elements C

Introduction such that A ≤ C ≤ B. Posets Depths Past Work Example Refinement The color blue represents the interval. Tools for finding ndepth Rotations H Classes Ndepth of a refinement F G -ordered poset

Results C D E Conclusion

Acknowledgments A B

An interval I = [A, G] Depths of Posets Ordered by Interval partitions of a poset Reﬁnement

Ying Gao Mentored by Sergei Bernstein • In an interval partition, the poset is completely partitioned into non-overlapping intervals. Each element Introduction Posets of the poset is in exactly one interval. Depths Past Work Example Reﬁnement

Tools for ﬁnding ndepth H Rotations Classes Ndepth of a F G reﬁnement -ordered poset

Results C D E Conclusion

Acknowledgments A B

An interval partition P • We say that a level n contains all elements of depth n.

Example

H Level 4

F G Level 3

C D E Level 2

A B Level 1

Depths of Posets Ordered by Depth Reﬁnement

Ying Gao Mentored by Sergei Bernstein Definition The depth of an element X0 in a poset is defined to be the Introduction Posets maximum possible number of elements in a chain Depths X0 > X1 > X2 > ··· > Xn from X0 to the bottom of the Past Work poset. Refinement

Tools for ﬁnding ndepth Rotations Classes Ndepth of a reﬁnement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Depth Reﬁnement

Tools for ﬁnding ndepth • We say that a level n contains all elements of depth n. Rotations Classes Ndepth of a Example reﬁnement -ordered poset H Level 4 Results

Conclusion F G Level 3

Acknowledgments C D E Level 2

A B Level 1 Depths of Posets Ordered by nDepth Reﬁnement

Ying Gao Mentored by Sergei Bernstein For an interval I , ndepth[I ] = max(depth[X ]) over all

Introduction elements X in I . Posets Depths • For I = [A, B], ndepth[I ] = depth[B]. Past Work

Refinement H Tools for finding ndepth Rotations Classes F G Ndepth of a refinement -ordered poset C D E Results

Conclusion A B Acknowledgments

Interval I = [A, G]. ndepth[I ] = depth[G] = 3. Depths of Posets Ordered by nDepth Reﬁnement

Ying Gao Mentored by Sergei Bernstein For a partition P of a poset, ndepth[P] = min(ndepth[I ])

Introduction over all intervals I in P. Posets Depths • Which interval in the partition has the least depth? Past Work

Refinement H Tools for finding ndepth Rotations Classes F G Ndepth of a refinement -ordered poset C D E Results

Conclusion A B Acknowledgments

ndepth[P] = ndepth[B, E] = 2. Depths of Posets Ordered by nDepth Reﬁnement

Ying Gao Mentored by Sergei Bernstein For a poset G, ndepth[G] = max(ndepth[P]) over all Introduction possible partitions P of G. Posets Depths • Which partition(s) of the poset have the greatest depth? Past Work

Reﬁnement

Tools for finding H H ndepth Rotations F G F G Classes Ndepth of a C D E refinement C D E -ordered poset A B A B Results Conclusion ndepth = 2 ndepth = 3 Acknowledgments ndepth[G] = 3. Depths of Posets Ordered by Past work Refinement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths Yinghui Wang: Past Work k Refinement • ndepth of a product of chains n \ 0 is (n − 1)dk/2e Tools for finding ndepth Biro, Howard, Keller, Trotter, and Young Rotations Classes • For a poset B of the non-empty subsets of an n-element Ndepth of a set ordered by inclusion, ndepth[B] ≥ n/2 refinement -ordered poset We will study the properties of posets comprised of Results partitions of sets ordered by refinement. Conclusion

Acknowledgments Depths of Posets Ordered by Set-Partitions Reﬁnement

Ying Gao Mentored by Sergei Bernstein • A partition of a set is the division of the set of distinct Introduction Posets points into subsets. Depths Past Work • Every element of the set is partitioned into some subset,

Reﬁnement and no element is within two or more subsets.

Tools for ﬁnding ndepth Examples Rotations Classes Ndepth of a reﬁnement -ordered poset

Results

Conclusion

Acknowledgments Partitions of a 6-element set. Depths of Posets Ordered by Reﬁnement Reﬁnement

Ying Gao Mentored by Sergei Bernstein • It is possible to order partitions of a set by refinement. Introduction • A partition of a set P is considered finer than another Posets b Depths partition Pa if all subsets within Pb are within some Past Work subset in Pa. If Pb is finer than Pa, Pa is coarser than Refinement Pb. Tools for finding ndepth Rotations Classes Examples Ndepth of a refinement -ordered poset

Results

Conclusion

Acknowledgments

Right partition is finer than left partition Depths of Posets Ordered by Posets of the refinement ordering Refinement

Ying Gao Mentored by • These posets depend solely on the size of the set that is Sergei Bernstein partitioned Introduction Posets Let G denote a poset ordered by reﬁnement of all set Depths i Past Work partitions of the set with i elements except the ”empty

Refinement partition”, the partition of each element into a separate Tools for finding subset. ndepth Rotations Classes Example Ndepth of a refinement -ordered poset

Results

Conclusion

Acknowledgments

G3. Depths of Posets Ordered by Reﬁnement

Ying Gao Mentored by Now that we know what a poset ordered by reﬁnement looks Sergei Bernstein like . . . Introduction Posets Depths Past Work

Reﬁnement

Tools for finding ndepth How do we find the ndepth of such a poset? Rotations Classes Ndepth of a refinement -ordered poset

Results Conclusion First, we need a few tools. Acknowledgments Examples

Depths of Posets Ordered by Rotations Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction • A rotation of a certain partition is any other partition Posets that may be obtained by rotating the partition around a Depths Past Work circle. Refinement • A rotation of an interval is the interval formed by Tools for finding rotating each element of the interval the same number ndepth Rotations of times. Classes Ndepth of a refinement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Rotations Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Results

Conclusion

Acknowledgments Examples

Depths of Posets Ordered by Classes of set partitions Reﬁnement

Ying Gao Mentored by Sergei Bernstein • It is possible to group set partitions into classes based on the sizes of subsets in the set partitions. Introduction Posets • The class C = (S1, S2, S3,..., Sn) with all Si positive Depths Past Work integers and S1 ≥ S2 ≥ S3 ≥ ... ≥ Sn includes all

Refinement partitions which consist of exactly n subsets of size Tools for finding S1, S2,..., Sn. ndepth Rotations • All rotations of a partition are in the same class as the Classes partition. Ndepth of a refinement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Classes of set partitions Reﬁnement

Conclusion

Acknowledgments Depths of Posets Ordered by ndepth[G5] Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets A Depths Past Work

Refinement (A) x1 (B) x5 (C) x10 B C [5] [4,1] [3,2] Tools for finding ndepth Rotations Classes Ndepth of a D E refinement -ordered poset (D) x15 (E) x10 (F) x10 Results [2,2,1] [3,1,1] [2,1,1,1] F Conclusion

Acknowledgments Poset of classes in G5 Lemma ndepth[G5] < 4.

Depths of Posets Ordered by ndepth[G5] Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths In order for it to be true that ndepth[G5] = L ... Past Work Refinement • It must be impossible to partition the poset so that all Tools for finding intervals I have ndepth[I ] > L ndepth Rotations Classes • It must be possible make a partiton P of G5 such that Ndepth of a each interval I in P has ndepth[I ] ≥ L refinement -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by ndepth[G5] Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Results Lemma

Conclusion ndepth[G5] < 4.

Acknowledgments Depths of Posets Ordered by ndepth[G5] Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets A Depths Past Work

Acknowledgments Poset of classes in G5 Depths of Posets Ordered by ndepth[G5] Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths • Past Work We attempt to show that ndepth[G5] = 3 by

Refinement constructing a partition P of G5 with ndepth 3. Tools for finding • It is possible to make a partition P with ten ndepth Rotations non-overlapping intervals of the form [F , C] and five Classes non-overlapping intervals of the form [D, B], as well as Ndepth of a refinement an interval [A, A], which will completely partition the -ordered poset

Results poset into intervals of ndepth 3 or greater. Conclusion Theorem Acknowledgments The ndepth of G5 is 3. Depths of Posets Ordered by ndepth[G5] Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction C C Posets Depths B Past Work

Reﬁnement

Tools for ﬁnding ndepth D E D E Rotations Classes

Ndepth of a D reﬁnement -ordered poset

Results

Conclusion F F Acknowledgments The 10 intervals of the form [F , C] and 5 intervals of the form [D, B] include all the rotations of these three intervals. • ndepth[Gi ] is non-decreasing; for all i, ndepth[Gi+1] ≥ ndepth[Gi ]

• ndepth[Gi ] does not increase very fast; for all i, ndepth[Gi+1] ≤ ndepth[Gi ] + 1

• ndepth[Gi ] increases inﬁnitely; for any ndepth L, there is some i large enough that ndepth[Gi ] = L.

Depths of Posets Ordered by Results Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets The values of ndepth[Gi ] Depths Past Work Refinement i 1 2 3 4 5 6 Tools for finding ndepth ndepth[Gi ] - 1 1 2 3 3 Rotations Classes Ndepth of a refinement -ordered poset

Results

Conclusion

Acknowledgments • ndepth[Gi ] does not increase very fast; for all i, ndepth[Gi+1] ≤ ndepth[Gi ] + 1

• ndepth[Gi ] increases inﬁnitely; for any ndepth L, there is some i large enough that ndepth[Gi ] = L.

Depths of Posets Ordered by Results Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Results

Conclusion

Acknowledgments • ndepth[Gi ] increases inﬁnitely; for any ndepth L, there is some i large enough that ndepth[Gi ] = L.

Depths of Posets Ordered by Results Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Acknowledgments Depths of Posets Ordered by Results Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets The values of ndepth[Gi ] Depths Past Work Refinement i 1 2 3 4 5 6 Tools for finding ndepth ndepth[Gi ] - 1 1 2 3 3 Rotations Classes • ndepth[Gi ] is non-decreasing; for all i, Ndepth of a refinement ndepth[Gi+1] ≥ ndepth[Gi ] -ordered poset • ndepth[Gi ] does not increase very fast; for all i, Results ndepth[G ] ≤ ndepth[G ] + 1 Conclusion i+1 i Acknowledgments • ndepth[Gi ] increases infinitely; for any ndepth L, there is some i large enough that ndepth[Gi ] = L. Depths of Posets Ordered by Future Work Refinement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths Past Work

Refinement • Describe upper and lower bounds for ndepth[Gi ] for Tools for finding ndepth very large i Rotations Classes • Prove that ndepth[Gi ] increases linearly Ndepth of a refinement • Concretely describe the sequence ndepth[Gi ] -ordered poset

Results

Conclusion

Acknowledgments Depths of Posets Ordered by Acknowledgments Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths Thanks to Past Work

Reﬁnement • My mentor, Sergei Bernstein, for his patience and

Tools for ﬁnding guidance ndepth Rotations • Professor Richard Stanley, for suggesting the project Classes Ndepth of a • Tanya Khovanova and Dai Yang, for their advice reﬁnement -ordered poset • PRIMES, for providing this research opportunity Results • My parents, for their continuing support Conclusion

Acknowledgments Depths of Posets Ordered by Bibliography Reﬁnement

Ying Gao Mentored by Sergei Bernstein

Introduction Posets Depths • Wang, Yinghui. (2010). ”A New Depth Related to the Past Work Stanley Depth of Some Power Sets of Multisets.” Reﬁnement

Tools for ﬁnding Online. http://arxiv.org/pdf/0908.3699v4.pdf. ndepth Rotations • Biro, C. Howard, D. Keller, M. Trotter, W. Young, S. Classes ”Interval Partitions and Stanley Depth.” Online. Ndepth of a reﬁnement http://rellek.net/home/images/publications/partition.pdf. -ordered poset

Results • Stanley, Richard. (2011). Enumerative Combinatorics,

Conclusion Volume 1, Second Edition, 277-284.

Acknowledgments