Four Variations on Graded Posets

We explore the enumeration of some natural classes of graded posets, including all graded posets, (2+2)- and (3+1)-avoiding graded posets, (2+2)-avoiding graded posets, and (3+1)-avoiding graded posets. We obtain enumerative and structural theorems for all of them. Along the way, we discuss a situation when we can switch between enumeration of labeled and unlabeled objects with ease, generalize a result of Postnikov and Stanley from the theory of hyperplane arrangements, answer a question posed by Stanley, and see an old result of Klarner in a new light.Comment: 28 page

Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology

Given any polytope $P$ and any generic linear functional ${\bf c}$, one obtains a directed graph $G(P,{\bf c})$ by taking the 1-skeleton of $P$ and orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. This paper raises the question of finding sufficient conditions on a polytope $P$ and generic cost vector ${\bf c}$ so that the graph $G(P, {\bf c} )$ will not have any directed paths which revisit any face of $P$ after departing from that face. This is in a sense equivalent to the question of finding conditions on $P$ and ${\bf c}$ under which the simplex method for linear programming will be efficient under all choices of pivot rules. Conditions on $P$ and ${\bf c}$ are given which provably yield a corollary of the desired face nonrevisiting property and which are conjectured to give the desired property itself. This conjecture is proven for 3-polytopes and for spindles having the two distinguished vertices as source and sink; this shows that known counterexamples to the Hirsch Conjecture will not provide counterexamples to this conjecture. A part of the proposed set of conditions is that $G(P, {\bf c} )$ be the Hasse diagram of a partially ordered set, which is equivalent to requiring non revisiting of 1-dimensional faces. This opens the door to the usage of poset-theoretic techniques. This work also leads to a result for simple polytopes in which $G(P, {\bf c})$ is the Hasse diagram of a lattice L that the order complex of each open interval in L is homotopy equivalent to a ball or a sphere of some dimension. Applications are given to the weak Bruhat order, the Tamari lattice, and more generally to the Cambrian lattices, using realizations of the Hasse diagrams of these posets as 1-skeleta of permutahedra, associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition substantially improved throughou

H_2-Optimal Decentralized Control over Posets: A State-Space Solution for State-Feedback

We develop a complete state-space solution to H_2-optimal decentralized control of poset-causal systems with state-feedback. Our solution is based on the exploitation of a key separability property of the problem, that enables an efficient computation of the optimal controller by solving a small number of uncoupled standard Riccati equations. Our approach gives important insight into the structure of optimal controllers, such as controller degree bounds that depend on the structure of the poset. A novel element in our state-space characterization of the controller is a remarkable pair of transfer functions, that belong to the incidence algebra of the poset, are inverses of each other, and are intimately related to prediction of the state along the different paths on the poset. The results are illustrated by a numerical example.Comment: 39 pages, 2 figures, submitted to IEEE Transactions on Automatic Contro

Adinkras for Mathematicians

Adinkras are graphical tools created for the study of representations in supersymmetry. Besides having inherent interest for physicists, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computational nature. We use a more mathematically natural language to survey these topics, suggest new definitions, and present original results.Comment: 34 pages, 14 figure