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
n = 4 nonempty subsets
n = 4 nonempty subsets
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}}
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}}
{ } {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}
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}}
{ } {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}
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}}
{3, 4}
{3, 4, 6} {3, 4, 7} {3, 4, 8}
A
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}}
{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 6}
{3, 4}
{3, 4, 6} {3, 4, 7} {3, 4, 8}
A
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}}
{3, 4, 6, 7} {3, 4, 6, 8} {3, 4, 7, 6}
{3, 4}
{3, 4, 6} {3, 4, 7} {3, 4, 8}
A
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}}
{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}
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}}
{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}
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}}
{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}
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}}
{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
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}}
{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
Randomized Algorithm
Collision
Randomized Algorithm
Backtracking Algorithm
Backtracking Algorithm
After every pick, check to see if there is a collision
Backtracking Algorithm
Collision
Backtracking Algorithm
If there is a collision, pick another ball.
Backtracking Algorithm
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.
1 - 4 config. 1 - 5 config.
n = 14
1 - 2 config. 1 – 3 config.
1 - 4 config. 1 - 5 config. 1 - 6 config.
1 - 7 config. 1 - 8 config.
Acknowledgments
• Professor Mitchel T. Keller • Summer Research Scholars Program • W&L Mathematics Department