the notion of a $d$-poset was introduced in a connection withquantum mechanical models. in this paper, we introduce theconditional expectation of  random variables on thek^{o}pka's $d$-poset and prove the basic properties ofconditional expectation on this  structure.

A binary poset code of codimension m (of cardinality 2n−m , where n is the code length) can correct maximum m errors. All possible poset metrics that allow codes of codimension m to be m-, (m − 1)-, or (m − 2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of r-perfect poset co...

We study a poset N on the free monoid X∗ on a countable alphabet X. This poset is determined by the fact that its total extensions are precisely the standard term orders on X ∗. We also investigate the poset classifying degree-compatible standard term orders, and the poset classifying sorted term orders. For the latter poset, we give a Galois coconnection with the Young lattice.

The interval poset of a permutation catalogues the intervals that appear in its one-line notation, according to set inclusion. We study this poset, describing structural, characterizing, and enumerative properties.

In this paper all poset structures are classi£ed which admit the MacWilliams identity, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. We prove that being a hierarchical poset is a necessary and suf£cient condition for a poset to admit MacWilliams identity. An explicit relation is also derived between P-weight distribution of a hierarchical ...

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A nondegenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial...