Liam Solus
Postdoctoral Researcher in Mathematics
KTH Royal Institute of Technology
Research Interests:
My research interests are in the field of combinatorics/discrete mathematics and its applications to algebra, geometry, statistics, and artificial intelligence.
Within each of the latter fields, researchers can often phrase important questions in terms of the properties of statistical distributions associated to algebraic, geometric, and/or discrete objects.
In recent years, my research has centered around developing and utilizing combinatorial methods to describe the relevant properties of such distributions.
Buzzwords related to my work include: algebraic and geometric combinatorics, lattice polytopes, convex and discrete geometry, AI, machine learning, causal inference, causality, graphical models, and algebraic statistics.
Professional Information:
I am currently an NSF Mathematical Sciences Postdoctoral Research Fellow at KTH.
NSF Postdoctoral Mentor: Dr. Petter Brändén.
In 2016 I was a postdoctoral researcher working at IST Austria and MIT.
Postdoctoral Mentor: Dr. Caroline Uhler.
In December 2015 I completed my PhD in mathematics at University of Kentucky.
Doctoral Advisor: Dr. Ben Braun.
In April 2014 I completed my masters in mathematics at University of Kentucky.
Masters Advisor: Dr. Ben Braun.
In May 2011 I completed my bachelors in mathematics at Oberlin College.
Honors Advisor: Dr. Kevin Woods.
Academic Advisor: Dr. Jim Walsh.
Publications:
 P. Brändén and L. Solus.
Symmetric decompositions and realrootedness.
Preprint (2018).
arXiv version available here.
 L. Solus.
Local h*polynomials of some weighted projective spaces.
To appear in the proceedings of The 2018 Summer Workshop on Lattice Polytopes, Osaka University (2018).
arXiv version available here.
 N. Gustafsson and L. Solus.
Derangements, Ehrhart theory, and local hpolynomials.
Submitted (2018).
arXiv version available here.
 F. Liu and L. Solus.
On the relationship between Ehrhart unimodality and Ehrhart positivity.
Submitted (2018).
arXiv version available here.
 E. Perrone, L. Solus, and C. Uhler.
Geometry of Discrete Copulas.
Submitted (2018).
arXiv version available here.
 L. Solus, Y. Wang, L. Matejovicova, and C. Uhler.
Consistency guarantees for greedy permutationbased causal inference algorithms.
Submitted (2018).
arXiv version available here.
 B. Braun, R. Davis, and L. Solus.
Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices.
Advances in Applied Mathematics. 100 (2018): 122142.
arXiv version available here.
 B. Braun and L. Solus.
rstable hypersimplices.
Journal of Combinatorial Theory, Series A 157 (2018) 349388.
arXiv version available here.
 A. Radhakrishnan, L. Solus, and C. Uhler.
Counting Markov equivalence classes for DAG models on trees.
Discrete Applied Mathematics.
DOI: https://doi.org/10.1016/j.dam.2018.03.015 (2018).
arXiv version available here.
 L. Solus.
Simplices for numeral systems.
Transactions of the American Mathematical Society.
DOI: https://doi.org/10.1090/tran/7424 (2017).
arXiv version available here.
 Y. Wang, L. Solus, K. Dai Yang, and C. Uhler.
Permutationbased causal inference algorithms with interventions.
Advances in Neural Information Processing Systems (2017).
arXiv version available here.
 A. Radhakrishnan, L. Solus, and C. Uhler.
Counting Markov equivalence classes by number of immoralities.
The Proceedings of the 2017 Conference on Uncertainty in Artificial Intelligence and The Proceedings of the 2017 UAI Special Workshop on Causality (2017).
arXiv version available here.
 V. Karwa, D. Pati, S. Petrovic, L. Solus, N. Alexeev, M. Raic, D. Wilburne, R. Williams, and B. Yan.
Exact tests for stochastic block models.
Submitted (2017).
arXiv version available here.
 L. Solus, C. Uhler, and R. Yoshida.
Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K_5 Minors.
Linear Algebra and its Applications (2016).
DOI 10.1016/j.laa.2016.07.026
arXiv version available here.
 T. Hibi and L. Solus.
The facets of the rstable (n,k)hypersimplex.
Annals of Combinatorics (2016).
DOI 10.1007/s000260160325x
arXiv version available here.
 J. Calcut, J. MetcalfBurton, T. Richard, L. Solus.
Borromean rays and hyperplanes.
The Journal of Knot Theory and Its Ramifications. Vol. 23, No. 4 (2014),
arXiv version available here.
 L. Solus.
A topological generalization of partition regularity.
Involve: A Journal of Mathematics 34 (2010), 421433. DOI 10.2140/involve.2010.3.421.
Available here.
 L. Solus.
Normal and ΔNormal Configurations in Toric Algebra.
Oberlin College Honors Theses, Bachelor of Arts, 2011.
Available here.
Students:
PhD Students:

Petter Restadh.
KTH Royal Institute of Technology.
Thesis topic: Combinatorics in Causality.
coadvising with Dr. Svante Linusson.
Dates: August 2018  present.
Masters Students:

Jonas Bederoff Eriksson.
KTH Royal Institute of Technology.
Thesis Title: Graph Properties of DAG Associahedra and Related Polytopes.
Dates: Completed June 2018.

Nils Gustafsson.
KTH Royal Institute of Technology.
Thesis Title: Box Polynomials of Lattice Simplices.
Dates: Completed June 2018.
Teaching Responsibilities:
This semester I am not teaching any courses.
However, in the previous semesters I had the privilege of teaching the following:
Primary Instructor:
MA 114, Calculus II, University of Kentucky (Spring 2015).
MA 123, Elementary Calculus and its Applications, University of Kentucky (Summer 2013)
Teaching Assistant:
MA 113, Calculus I, University of Kentucky (Fall 2015).
MA 114, Calculus II, University of Kentucky (Spring 2015).
MA 138, Calculus II for the Life Sciences, University of Kentucky (Spring 2014).
MA 137, Calculus I for the Life Sciences, University of Kentucky (Fall 2014, Fall 2013).
MA 123, Elementary Calculus and its Applications, University of Kentucky (Spring 2013, Fall 2012).
Some Combinatorial Sequences I Like and Why I Like Them:
 OEIS ID: A000045, The Fibonacci Numbers.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040,...
Defined by the recursion F(0):=0, F(1):=1, and F(n) :=F(n1)+F(n2) for n>1, the Fibonacci numbers are perhaps one of the most ubiquitious combinatorial sequences in mathematics and its applications. They appear in our calculus classes, and even sometimes on our pineapples! So it is always a treat when the Fibonacci numbers make an appearance in your work. Most recently, this happened to me (along with A. Radhakrishnan and C. Uhler) while studying the number of Markov equivalence classes (MECs) of directed acyclic graphical (DAG) models whose underlying undirected graph is a path (see here). Every directed acyclic graph (DAG) encodes a set of conditional independence (CI) statements meant to capture causal relations existing between the random variables associated to the nodes. The undirected edges encode dependencies and the directions of the arrow can be selected (through interventional methods) to encode causal implications. Without interventional testing, two DAG models may be indistinguishable up to their encoded CI relations, and thus belong to the same Markov equivalence class (MEC). By restricting the structure of the skeleton and counting these MECs we can uncover some beautiful combinatorial sequences hidden away in causal inference and AI research!
 OEIS ID: A011973, The Fibonacci Polynomials.
0,
1,
1,
1, 1,
1, 2,
1, 3, 1,
1, 4, 3,
1, 5, 6, 1,
1, 6, 10, 4,
1, 7, 15, 10, 1,
1, 8, 21, 20, 5,
1, 9, 28, 35, 15, 1,
1, 10, 36, 56, 35, 6,
1, 11, 45, 84, 70, 21, 1,
1, 12, 55, 120, 126, 56, 7,....
The nth row in the above triangle is the coefficient vector of a polynomial called the nth Fibonacci polynomial. Notice that summing the entries in each row returns the nth Fibonacci number. So it is not surprising that the Fibonacci polynomials often offer combinatorial refinements when we find objects enumerated by the Fibonacci numbers. Continuing our story from causality and AI, the kth coefficient of the nth Fibonacci polynomial counts the number of MECs on the path with n nodes having k immoralities; i.e. having k of a special type of induced subgraph that serves to characterize the Markov equivalence class of a given DAG on a fixed underlying undirected graph (see here). Classic results in combinatorics tell us that the nth Fibonacci polynomial has only realroots, and as a consequence, the (discrete) distribution encoding the number of immoralities in a MEC on the path tends to the normal distribution as n tends to infinity.
 OEIS ID: A318407, The MECPolynomial of Caterpillar Graphs.
0,
1,
1,
1, 1,
1, 2,
1, 4, 1, 1,
1, 5, 3, 1,
1, 7, 8, 3, 3,
1, 8, 13, 6, 4,
1, 10, 23, 16, 13, 6, 1,
1, 11, 31, 29, 19, 10, 1,
1, 13, 46, 59, 46, 39, 13, 5,
1, 14, 57, 90, 75, 58, 23, 6,
1, 16, 77, 153, 158, 147, 97, 39, 15, 1,
1, 17, 91, 210, 248, 222, 155, 62, 21, 1,...
A close relative to the path graph is the caterpillar graph. When we count MECs on caterpillar graphs by their number of immoralities, we get out polynomials whose coefficient vectors are the rows in the above triangle. These polynomials admit a recursion akin to that of the Fibonacci polynomials (see here). However, we don't get the nice realrooted property as with the path. At the same time, by inspection, it appears that these polynomials are always unimodal; that is, the coefficient vector of each polynomial is a unimodal sequence. It would be interesting to see a proof of this claim!
 OEIS ID: A318405, The Number of MECs on a Spider Graph.
1, 2, 1, 5, 5, 1, 13, 15, 12, 1, 34, 71, 49, 27, 1, 89, 287, 409, 163, 58, 1, 233, 1237, 2596, 2315, 537, 121, 1, 610, 5205, 18321, 23393, 12709, 1739, 248, 1, 1597, 22105, 124177, 268893, 205894, 67919, 5537, 503, 1,...
The above sequence is the rectangular array of numbers R(n,k):=F(n+1)^knF(n1)F(n)^{k1} read by upwards antidiagonals, where F(n) denotes the nth Fibonacci number; n>0, k>1. The number R(n,k) counts the number of MECs on the spider graph with k legs, each of length n. This sequence generalizes the wellknown identity F(2n) = F(n+1)^22F(n1)F(n), and at the same time further highlights the Fibonaccirelated combinatorial sequences hidden within the field of causal inference and AI.
 OEIS ID: A007984, The Number of MECs for DAG Models on n Nodes.
1, 2, 11, 185, 8782, 1067825, 312510571, 212133402500, 326266056291213, 1118902054495975181, 8455790399687227104576, 139537050182278289405732939, 4991058955493997577840793161279,...
Perhaps the most important sequence of numbers pertaining to DAG models is this one. The nth term in this sequence enumerates the number of MECs on n nodes over all directed acyclic graphs (i.e. not restricting to a specific underlying undirected graph as in our previous examples). The key observation to be made about this sequence is that it grows superexponentially in n and on the order of the number of DAGs on n nodes. In biology, AI, and the social sciences, researchers would like to have efficient algorithms for learning a DAG model based only on observational data. The observed growth rate of this sequence shows us that we cannot simply consider each MEC and pick the best one. Instead, we need to devise tricky algorithms that work efficiently and reliably through this very large search space. This is one of the fundamental challenges facing researchers in causal inference/causality.
 OEIS ID: A008292, The Eulerian Polynomials.
1,
1, 1,
1, 4, 1,
1, 11, 11, 1,
1, 26, 66, 26, 1,
1, 57, 302, 302, 57, 1,
1, 120, 1191, 2416, 1191, 120, 1,
1, 247, 4293, 15619, 15619, 4293, 247, 1,...
Similar to the Fibonacci numbers, the Eulerian polynomials are ubiquitous throughout algebraic and geometric combinatorics. The kth coefficient of the nth Eulerian polynomial counts the number of permutations of [n] with exactly k descents. The coefficient vectors of these polynomials satisfy all of combinatorists' favorite distributional properties: they are realrooted, symmetric, logconcave, and unimodal. It is always exciting when you discover a new interpretation of the Eulerian polynomials. Most recently, this happened to me when I found that the nth Eulerian polynomial is the Ehrhart h*polynomial of a curious simplex called the factoradic nsimplex (see here).
 OEIS ID: A046739, The Derangement Polynomials.
0,
1,
1, 1,
1, 7, 1,
1, 21, 21, 1,
1, 51, 161, 51, 1,
1, 113, 813, 813, 113, 1,
1, 239, 3361, 7631, 3361, 239, 1,
1, 493, 12421, 53833, 53833, 12421, 493, 1,
1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1,...
A close relative of the Eulerian polynomials are the derangement polynomials.
In fact, the derangement polynomials are examples of restricted Eulerian polynomials, in which we still enumerate by number of descents in permutations of [n] but restrict ourselves to only a subset of the permutations. In this case, we consider only those permutations that are derangements; i.e., permutations with no fixed points. When the Eulerian polynomials make an appearance in algebraic/geometric combinatorics, the derangement polynomials are often lurking nearby. This is seen in the study of hpolynomials for subdivisions of simplicial complexes: the hpolynomial of the barycentric subdivision of a simplex is the nth Eulerian polynomial, and the local hpolynomial of the subdivision is the nth derangement polynomial. In a parallel story within algebraic combinatorics, N. Gustafsson and I recently discovered that the slecture hall simplex with s = (2,3,...,n+1) has Ehrhart h*polynomial the nth Eulerian polynomial and local h*polynomial the nth derangement polynomial (see here).
 OEIS ID: A130477, The Maximum Descent Polynomials.
1,
1, 1,
1, 2, 3,
1, 3, 8, 12,
1, 4, 15, 40, 60,
1, 5, 24, 90, 240, 360,
1, 6, 35, 168, 630, 1680, 2520,
1, 7, 48, 280, 1344, 5040, 13440, 20160,
1, 8, 63, 432, 2520, 12096, 45360, 120960, 181440,...
An intriguing and lessstudied relative of the Eulerian polynomials is another polynomial enumerating a descenttype statistic. In this case, the kth entry in the nth row of this triangle is the number of permutations of [n] with maximum descent being k (assuming that the identity permutation has maximum descent 0). In a recent paper, we found that when the kth entry of the nth row is taken as the weight of coordinate x_k for k = 0,...,n1 in an (n1)dimensional weighted projective space, the result is the toric variety defined by an ndimensional simplex whose Ehrhart h*polynomial is the nth Eulerian polynomial  a surprising geometric interpolation between these two descenttype statistics!
 OEIS ID: A318408, The Local h*Polynomial of the Factoradic Simplices.
0,
0,
1,
1, 1,
1, 6, 1,
1, 19, 19, 1,
1, 48, 142, 48, 1,
1, 109, 730, 730, 109, 1,
1, 234, 3087, 6796, 3087, 234, 1,
1, 487, 11637, 48355, 48355, 11637, 487, 1,
1, 996, 40804, 291484, 543030, 291484, 40804, 996, 1,...
Like our previous examples, the nth row in this triangle is the coefficient vector of a polynomial that enumerates descents of specially chosen permutations of [n]; i.e. of a restricted Eulerian polynomial. In this case, we first order the collection of all permutations of [n] in lexicographic order from smallesttolargest, indexing them starting with 0. The desired polynomial then enumerates the number of descents in those permutations whose index in this list is congruent to 1 or 5 modulo 6. The result is a realrooted, symmetric, and unimodal polynomial admitting a geometric interpretation studied here.
 The γVector of the Local h*Polynomial of the Factoradic Simplices.
0,
0,
1,
1,
1, 4,
1, 16,
1, 44, 48,
1, 104, 408,
1, 228, 2160, 1088,
1, 480, 9216, 15872,
1, 988, 34848, 137216, 39680,...
Classic results tell us that if a symmetric polynomial with nonnegative coefficients is also realrooted then it will have nonnegative coefficients when we change to another polynomial basis, called the γbasis. The local h*polynomials of the factoradic simplices, discussed in our previous example, have all of these properties! We see here the coefficients of these local h*polynomials in the γbasis. Unlike in the standard basis, we do not yet know of any combinatorial interpretation of these numbers. Try and find one!
Personal Interests:
Aside from mathematics, I like to surf, snowboard, climb, exercise, hike, travel, and play my violin.
Liam Solus
Postdoctoral Researcher in Mathematics
KTH Royal Institute of Technology
Last updated on 7 September, 2018