At IISc Bangalore, I have started studying the computation complexity of cube representations of trees. We believe that finding the minimum dimension on which trees can be represented as the intersection graphs of unit hypercubes is NP-complete. This is supported by our most recent result that a vertex contact version of cubicity is NP-complete even for trees. This result also shows that embedding trees as induced subgraphs of the 2-dimensional grid graph is NP-complete. We give a constant factor approximation scheme for computing the vertex contact cubicity of trees. 

Update: We have found out that the NP-hardness of the vertex contact version of cubicity for trees was already known since 2004. We have proven now that segment contact versions and the recognition problem of whether cubicity is 2 or not for trees are also NP-complete.

At IIT Bombay, I started working on algebraic properties of polynomials arising in Enumerative Combinatorics. We studied gamma-positivity, log-concavity and real rootedness of polynomials that came up in permutation enumeration. I was able to prove the polynomials that enumerated descents and excedances over the even permutations had ultra-log-concave coefficients. It is conjectured that these polynomials have only real roots. I was also able to give a criterion that certifies log-concavity of combinatorial sequences that satisfy triangular recurrences. This criterion gives a unified approach to log-concavity of many sequences that appear in combinatorics such as the Eulerian numbers, Stirling numbers and Lah numbers. 

Update: The study had let me construct a family of polynomials which has been dubbed the 'Super'-Eulerian polynomials by Prof. Per Alexandersson. I love the name. Thanks, professor. Prof. Alexandersson has proven that these polynomials are real-rooted and gamma-positive after a conjecture of mine. I was independently studying this conjecture from a multivariate stability angle and was able to extend his results to real-stability of an associated multivariate generalisation of the polynomial. The work now gives a unified approach to real-rootedness of many enumerative combinatorial polynomials such as the ones associated with the Eulerian, Stirling and Lah numbers. We have also given a variety of different combinatorial models to explain these super-Eulerian numbers including a poset model that I conjecture is PECK for all pairs of parameters.

Later on in my study at IIT Bombay, I took to studying the pattern avoidance in permutations. We were able to reformulate some recent results of Burstein, Kitaev, Han and Zhang in the language of sums of labelled posets which allowed us to ask more questions from a poset perspective. We were able to completely classify the shape-Wilf equivalence of poset patterns of length upto 5 whose connected components were chains. We also gave a new insertion encoding and bijection that we used to completely classify shape-Wilf equivalence of sets of short patterns.