A differential poset is a partially ordered set with raising and lowering operators U and D which satisfy the commutation relation DU−UD = rI for some constant r. This notion may be generalized to deal with the case in which there exist sequences of constants {qn}n≥0 and {rn}n≥0 such that for any poset element x of rank n, DU(x) = qnUD(x)+rnx. Here, we introduce natural raising and lowering ope...

In this paper we study quotients of posets by group actions. In order to define the quotient correctly we enlarge the considered class of categories from posets to loopfree categories: categories without nontrivial automorphisms and inverses. We view group actions as certain functors and define the quotients as colimits of these functors. The advantage of this definition over studying the quoti...

We show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1+q. We extend this enumerative result via a decomposition of a new poset Π(n, k) which we call the Stirling poset of the second kind. Its rank generating function is the q-Stirling number S...

There has been much research on the combinatorial problem of generating the linear extensions of a given poset. This paper focuses on the reverse of that problem, where the input is a set of linear orders, and the goal is to construct a poset or set of posets that generates the input. Such a problem finds applications in computational neuroscience, systems biology, paleontology, and physical pl...

The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower bound of Ω( √ n) is shown for the queuenumber of the class of n-element planar posets. The queuenu...

Linear discrepancy and weak discrepancy have been studied as a measure of fairness in giving integer ranks to points of a poset. In linear discrepancy, the points are totally ordered, while in weak discrepancy, ties in rank are permitted. In this paper we study the t-discrepancy of a poset, a hybrid between linear and weak discrepancy, in which at most t points can receive the same rank. Intere...

Let P be a finite partially ordered set with unique minimal element 0̂. We study the Betti poset of P , created by deleting elements q ∈ P for which the open interval (0̂, q) is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form (0̂, p) ⊂ P and corresponding open intervals in the Betti poset. Our motivating application is that the...

The dimension of a poset P , denoted dim(P ), is the least positive integer d for which P is the intersection of d linear extensions of P . The maximum dimension of a poset P with |P | ≤ 2n+ 1 is n, provided n ≥ 2, and this inequality is tight when P contains the standard example Sn. However, there are posets with large dimension that do not contain the standard example S2. Moreover, for each f...

The characterization of Fibonacci Cobweb poset P as DAG and oDAG is given. The dim 2 poset such that its Hasse diagram coincide with digraf of P is constructed. 1 Fibonacci cobweb poset The Fibonacci cobweb poset P has been invented by A.K.Kwaśniewski in [1, 2, 3] for the purpose of finding combinatorial interpretation of fibonomial coefficients and eventually their reccurence relation. In [1] ...

Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that principal order filter at every vertex is isomorphic to whole poset. Well-known examples include k-ary trees, grid graphs, and Stern Very little known about upho in general. In this paper, we construct Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions h...