1. Aswasan Joshi
Stanley Depth of Monomial Ideals:
A Computational Investigation

2. 3x 5−πx 3+3x+ √7

3. Coefficients
3x 5−πx 3+3x+ √7

4. 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

5. 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 Integers⅚

6. 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.

7. K is the rational numbers
½x1
3x2 −3x3 +7x1
&
⅝x1
100 −5x3 +6
examples of polynomials in
K [x1,x2,x3]

8. Squarefree monomial ideals
• A monomial is called
squarefree if each ai is 0 or 1.
✓ x5x8x9 ✗ x5
4x8
2x9
• A monomial ideal is called squarefree if it
is generated by squarefree monomials.
Stanley Depth
!!
!!
!!
!!
⋯!!!
!!

9. Source: Richard P. Stanley

10. 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}

11. 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

12. 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

13. 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}

14. 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

15. n = 4
nonempty subsets
1 2 3
12 3123
123
4
14 24 34
124 134 234
1234

16. sdepth!! =!max
!
!!!sdepth!!
!!!!!!!!!!!!!!!!!!!=!max
!
min
!,!!!! ∈!
|!|
P – Poset
Q – Partition

17. 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.

18. What happens for
all subsets of {1, 2,…, n}
of size at least 2?
sdepth!! = ⌈! + 4
3
⌉!
On the Stanley depth of squarefree Veronese Ideals.
KELLER, M. T., SHEN, Y.-H., STREIB, N., AND YOUNG, S. J.
J. Alg. Combin. (2011)

19. 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}}

20. 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}

21. 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}

22. 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

23. 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

24. 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

25. 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}

26. 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}

27. 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}

28. 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

29. 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

30. 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

32. Randomized Algorithm

33. Randomized Algorithm
Collision

34. Randomized Algorithm

36. Backtracking Algorithm

37. Backtracking Algorithm
After every pick, check to see
if there is a collision

38. Backtracking Algorithm
Collision

39. Backtracking Algorithm
If there is a collision, pick another ball.

40. Backtracking Algorithm
After every pick check to see
if there is a collision

41. For
n
=
11
6.845 * 10105

42. 1078 – 1082
atoms
in the observable,
known universe

43. For
n
=
14
1.618 * 10245
6.247
*
10238
possible
par<<ons
per
second

44. 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

45. 1
2
3
4
5
6
7
8
A single rotation
clockwise
n = 8
[{2,3},{2,3,4,5}]

47. 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

48. n = 8
1 - 2 config. 1 - 3 config.
1 - 4 config. 1 - 5 config.

49. n = 14
1 - 2 config. 1 – 3 config.
1 - 4 config. 1 - 5 config. 1 - 6 config.
1 - 7 config. 1 - 8 config.

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