A linear extension of a partial order P is a linear order L = x1, x2..., xn respecting the order relations of P , i.e. xi < xj implies i < j for all xi, xj ∈ P . In other words, L is the linear sum L = C0 ⊕ C1 ⊕ . . .⊕ Cm of disjoint chains C0, C1, . . . , Cm in P , whose union is all of P , such that x ∈ Ci, x ′ ∈ Cj and x < x ′ implies i ≤ j. We may assume the chains to be maximal, i.e. the l...

First, Cogis and Habib (RAIRO Inform. 7Mor. 13 (1979), 3-18) solved the jump number problem for series-parallel partially ordered sets (posets) by applying the greedy algorithm and then Rival (Proc. Amer. Math. Sot. 89 (1983). 387-394) extended their result to N-free posets. The author (Order 1 (1984), 7-19) provided an interpretation of the latter result in the terms of arc diagrams of posets ...

T fastest long-period random number generators currently available are based on linear recurrences modulo 2. So far, software that provides multiple disjoint streams and substreams has not been available for these generators because of the lack of efficient jump-ahead facilities. In principle, it suffices to multiply the state (a k-bit vector) by an appropriate k × k binary matrix to find the n...

Hiroshi Haramoto, Makoto Matsumoto Department of Mathematics, Hiroshima University, Kagamiyama 1-3-1 Higashi-Hiroshima, Hiroshima 739-8526, Japan {[email protected] and [email protected]} Takuji Nishimura Department of Mathematical Sciences, Yamagata University, Yamagata 990-8586, Japan {[email protected]} François Panneton and Pierre L’Ecuyer Département ...

1) Introduction and notations In this first section we will give our main definitions and recall different characterizations of interval and semi-orders. In section 2 we shall prove that after a decomposition routine, semi-orders have at most 2 consecutive bumps in a linear extension. We also prove, using a "divide-and-conquer" argument, that computing polynomially the jump number can be done p...