SRS presentation - Stanley Depth

Aswasan Joshi

Stanley Depth of Monomial Ideals: A Computational Investigation

3x 5−πx 3+3x+ √7

Coefficients

3x 5−πx 3+3x+ √7

½

⅓

-⅔

¼ ¾

-⅕

⅖

⅗

⅘

⅙

⅚

⅛ -⅜ ⅝

½ ⅕ ⅘

⅓ ⅖ ⅘

½

⅜ ⅝

-⅜

⅝ ⅘ ⅔

-⅘

⅜

½ ⅓ ½

⅓ ⅘

-⅕

-⅘ -⅗

Field

Addition, Subtraction, Multiplication & Division by nonzero numbers in the collection are all defined

-⅝

-⅘

½

⅓

-⅔

¼ ¾

-⅕

⅖

⅗

⅘

⅙

⅛ -⅜ ⅝

½ ⅕ ⅘

⅓ ⅖ ⅘

½

⅜ ⅝

-⅜

⅝ ⅘ ⅔

-⅘

⅜

½ ⅓ ½

⅓ ⅘

-⅕

-⅘ -⅗

Field

Addition, Subtraction, Multiplication & Division by nonzero numbers in the collection are all defined

-⅝

-⅘ ✓ ✗ Rational, Real Complex numbers Integers ⅚

If K is a field, the polynomial ring in n variables,

denoted K [ x1,...,xn ], consists of all polynomials

where the coefficients come from the field K

and the variables x1,x2,...,xn are all allowed to appear.

K is the rational numbers

½x13x2 −3x3 +7x1

& ⅝x1

100 −5x3 +6

examples of polynomials in K [x1,x2,x3]

Squarefree monomial ideals •  A monomial is called

squarefree if each ai is 0 or 1. ✓ x5x8x9 ✗ x5

4x82x9

•  A monomial ideal is called squarefree if it is generated by squarefree monomials.

Stanley Depth

!!!!!!!!⋯ !!!!!

Source: Richard P. Stanley

How to compute the Stanley depth of a monomial ideal

HERZOG, J., VLADOIU, M., AND ZHENG, X

Journal of Algebra

11/2009 •  Possible to compute the Stanley depth of a

squarefree monomial ideal using only techniques from discrete mathematics

•  It is enough to look at some of the subsets of

{ x1, ..., xn }, which is equivalent to considering subsets of {1, 2, ..., n}

Our goal is then to partition this collection C of sets into intervals that do not collide and cover the

whole poset.

1 2 3 4

12 13 23 14 24 34

123 124 134 234

1234

n = 4 nonempty subsets

If A and B are subsets of {1, 2, . . . , n}, the interval [A, B] contains every set T such that

A is a subset of T and T is a subset of B. (We call B the upper bound of the interval.)

[ {1, 2}, {1, 2, 4, 6} ]

{1, 2}, {1, 2, 4}, {1, 2, 6}, {1, 2, 4, 6}

{1, 2, 5} and {2, 4, 6} is not in this interval

Two intervals collide if they have at least a set in common.

[{1, 6}, {1, 6, 2, 3}] [{1, 3}, {1, 3, 6, 9}]

{1, 3, 6}

Collision {1, 6} {1,6,2}

{1, 6, 2, 3} {1, 3}

{1,3,9} {1, 3, 6, 9}

Our goal is then to partition this collection C of sets into intervals that do not collide and cover the

whole poset.

1 2 3 4

12 13 23 14 24 34

123 124 134 234

1234



1 2 3

12 3123

123

4

14 24 34

124 134 234

1234

sdepth!! = !max!

!!!sdepth!!

!!!!!!!!!!!!!!!!!!!= !max!

min!,!!!! ∈!

|!| P – Poset

Q – Partition

Interval partitions and Stanley depth

BIRO ́ , CS., HOWARD, D. M., KELLER, M. T., TROTTER, W. T., AND YOUNG, S. J.

J. Combin. Theory Ser. A

5/2010

For the case where all nonempty subsets of {1,2,...,n} are considered, it is possible to find a partition in which every interval’s upper bound

has size at least n/2.

What happens for all subsets of {1, 2,…, n}

of size at least 2?

sdepth!! = ⌈! + 43 ⌉!On the Stanley depth of squarefree Veronese Ideals.

KELLER, M. T., SHEN, Y.-H., STREIB, N., AND YOUNG, S. J. J. Alg. Combin. (2011)

expand_interval Input: [{3, 4}, {3, 4, 6, 7, 8}]

Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}


Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}

{ } {3} {4} {6} {7} {8}

{3, 4} {3, 6} {3, 7} {3, 8} {4, 6}

{4, 7} {4, 8} {6, 7} {6, 8} {7, 8}

{3, 4, 6} {3, 4, 7} {3, 4, 8} {3, 6, 7} {3, 6, 8}

{3, 7, 8} {4, 6, 7} {4, 6, 8} {4, 7, 8} {6, 7, 8}

{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 8} {3, 6, 7, 8} {4, 6, 7, 8} {3,4,6,7,8}

Powerset of {3, 4, 6, 7, 8}


Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}

{3, 4}

{3, 4, 6} {3, 4, 7} {3, 4, 8}

A


Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}

{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 6}

{3, 4}

{3, 4, 6} {3, 4, 7} {3, 4, 8}

A


Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}

{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 6} {3, 4, 7, 8}

{3, 4}

{3, 4, 6} {3, 4, 7} {3, 4, 8}

A

{3, 4, 8, 6}


Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}

{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 6} {3, 4, 7, 8}

{3, 4}

{3, 4, 6} {3, 4, 7} {3, 4, 8}

A

{3, 4, 8, 6} {3, 4, 8, 7}


Output: {{3,4}, {3,4,6}, {3,4,7}, {3,4,8}, {3,4,6,7},

{3,4,6,8}, {3,4,7,8}, {3,4,6,7,8}}

{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 6} {3, 4, 7, 8}

{3, 4}

{3, 4, 6} {3, 4, 7} {3, 4, 8}

A

{3, 4, 8, 6} {3, 4, 8, 7}

{3, 4, 6, 7, 8} B

find_partition Input: n = 8

Output: [[[1, 2], [1, 2, 3, 4]], [[2, 3], [2, 3, 4, 5]], [[3, 4], [3, 4, 5, 6]], [[4, 5], [4, 5, 6, 7]], [[5, 6], [5, 6, 7, 8]], [[6, 7], [6, 7, 8, 1]], [[7, 8], [7, 8, 1, 2]], [[8, 1], [8, 1, 2, 3]], [[1, 3], [1, 3, 4, 5]], [[2, 4], [2, 4, 5, 6]], [[3, 5], [3, 5, 6, 7]], [[4, 6], [4, 6, 7, 8]], [[5, 7], [5, 7, 8, 1]], [[6, 8], [6, 8, 1, 2]], [[7, 1], [7, 1, 2, 3]], [[8, 2], [8, 2, 3, 4]], [[1, 4], [1, 4, 5, 6]], [[2, 5], [2, 5, 6, 7]], [[3, 6], [3, 6, 7, 8]], [[4, 7], [4, 7, 8, 1]], [[5, 8], [5, 8, 1, 2]], [[6, 1], [6, 1, 2, 3]], [[7, 2], [7, 2, 3, 4]], [[8, 3], [8, 3, 4, 5]], [[1, 5], [1, 5, 2, 6]], [[2, 6], [2, 6, 3, 7]],

[[3, 7], [3, 7, 4, 8]], [[4, 8], [4, 8, 5, 1]]]

•  No Collision •  Covers the whole Poset

Randomized Algorithm


Collision

Backtracking Algorithm


After every pick, check to see if there is a collision


Collision


If there is a collision, pick another ball.


After every pick check to see if there is a collision

For n = 11 6.845 * 10105

1078 – 1082

atoms

in the observable, known universe

For n = 14 1.618 * 10245

6.247 * 10238 possible par<<ons per second

Ring Diagram to make Intervals

1

2

3

4

5

6

7

8

[{1,2},{1,2,3,4}] [A, B]

n = 8

= A

+ = B

1

2

3

4

5

6

7

8

A single rotation clockwise

n = 8

[{2,3},{2,3,4,5}]

ring_to_interval Input: [1,1,2,2,0,0,0,0] Output: [[[1, 2], [1, 2, 3, 4]], [[2, 3], [2, 3, 4, 5]], [[3, 4],

[3, 4, 5, 6]], [[4, 5], [4, 5, 6, 7]], [[5, 6], [5, 6, 7, 8]], [[6, 7], [6, 7, 8, 1]], [[7, 8], [7, 8, 1, 2]], [[8, 1], [8, 1, 2,3]]]

1 1

2

2 0

0

0

0 1

2

3

4

5

6

7

8

n = 8

n = 8

1 - 2 config. 1 - 3 config.


n = 14

1 - 2 config. 1 – 3 config.

1 - 4 config. 1 - 5 config. 1 - 6 config.


Acknowledgments

•  Professor Mitchel T. Keller •  Summer Research Scholars Program • W&L Mathematics Department

SRS presentation - Stanley Depth

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