Algorithms for computing lengths of chains in integral partition lattices

AbstractLet Pl,n denote the partition lattice of l with n parts, ordered by Hardy–Littlewood–Polya majorization. For any two comparable elements x and y of Pl,n, we denote by M(x,y), m(x,y), f(x,y), and F(x,y), respectively, the sizes of four typical chains between x and y: the longest chain, the shortest chain, the lexicographic chain, and the counter-lexicographic chain. The covers u=(u1,…,un)≻v=(v1,…,vn) in Pl,n are of two types: N-shift (nearby shift) where vi=ui−1, vi+1=ui+1+1 for some i; and D-shift (distant shift) where ui−1=vi=vi+1=⋯=vj=uj+1 for some i and j. An N-shift (a D-shift) is pure if it is not a D-shift (an N-shift). We develop linear algorithms for calculating M(x,y), m(x,y), f(x,y), and F(x,y), using the leftmost pure N-shift first search, the rightmost pure D-shift first search, the leftmost N-shift first search, and the rightmost D-shift first search, respectively. Those algorithms have significant applications in complexity analysis of biological sequences

On Multidimensional Inequality in Partitions of Multisets

We study multidimensional inequality in partitions of finite multisets with thresholds. In such a setting, a Lorenz-like preorder, a family of functions preserving such a preorder, and a counterpart of the Pigou-Dalton transfers are defined, and a version of the celebrated Hardy-Littlewood-Pölya characterization results is provided.Multisets, majorization, Lorenz preorder, Hardy-Littlewood-Polya theorem, transfers

Strict partitions and discrete dynamical systems

AbstractWe prove that the set of partitions with distinct parts of a given positive integer under dominance ordering can be considered as a configuration space of a discrete dynamical model with two transition rules and with the initial configuration being the singleton partition. This allows us to characterize its lattice structure, fixed point, and longest chains as well as their length, using Chip Firing Game theory. Finally, we study the recursive structure of infinite extension of the lattice of strict partitions

Ancestor ideals of vector spaces of forms, and level algebras

Let R be the polynomial ring in r variables over a field k, with maximal ideal M and let V denote a vector subspace of the space of degree-j homogeneous elements of R. We study three related algebras determined by V. The first is the ``ancestor algebra'' whose defining ideal is the largest graded ideal whose intersection with M^j is the ideal (V). The second is the ``level algebra'', whose defining ideal L(V) is the largest graded ideal of R such that the degree-j component is V; and third is the algebra R/(V). When r=2, we determine the possible Hilbert functions H for each of these algebras, and as well the dimension of each Hilbert function stratum. We characterize the graded Betti numbers of these algebras in terms of certain partitions depending only on H, and give the codimension of each stratum in terms of invariants of the partitions. When r=2 and k is algebraically closed the Hilbert function strata for each of the three algebras satisfy a frontier property that the closure of a stratum is the union of more special strata. The family G(H) of all graded quotients of R having the given Hilbert function is a natural desingularization of this closure.Comment: 41 pages, LateX file, to appear Journal of Algebr

Notes on integer partitions

Some observations concerning the lattices of integer partitions are presented. We determine the size of the standard contexts, discuss a recursive construction and show that the lattices have unbounded breadth

Properties of four partial orders on standard Young tableaux

Let SYT_n be the set of all standard Young tableaux with n cells. After recalling the definitions of four partial orders, the weak, KL, geometric and chain orders on SYT_n and some of their crucial properties, we prove three main results: (i)Intervals in any of these four orders essentially describe the product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. (ii) The map sending a tableau to its descent set induces a homotopy equivalence of the proper parts of all of these orders on tableaux with that of the Boolean algebra 2^{[n-1]}. In particular, the M\"obius function of these orders on tableaux is (-1)^{n-3}. (iii) For two of the four orders, one can define a more general order on skew tableaux having fixed inner boundary, and similarly analyze their homotopy type and M\"obius function.Comment: 24 pages, 3 figure