β Scribed by David Magagnosc
No coin nor oath required. For personal study only.
In this paper, the author develops a general formnla for numbers which are extensions of the Eulerian numbers. A~'n, r) is defined to be the number of n-permutations with r rises of magnitude at least m. Intermediate steps in the derivation include a vertical recurrence ~elation. recurrence relations of the diagonals both as numbers and as polynomials, and recurrences and formulae strictly on the diagonals.
π SIMILAR VOLUMES
A digraph D with n vertices is said to be decomposable into a set S of dicycles if every arc of D is contained in exactly one member of S. Counterexamples are given to the following conjectures which are generalizations of three well-known conjectures by G. Hajos, P. Erdos, and P. J. Kelly: (1) [B.
Let R = nβ₯0 R n be a positively graded commutative Noetherian ring. A graded R-module is called \*indecomposable (respectively \*injective) if it is indecomposable (respectively injective) in the category of graded R-modules. Let M = nβ M n be a finitely generated graded R-module. The first main re
be two sequences of events, and let & N (A) and & M (B) be the number of those A i and B j , respectively, that occur. We prove that Bonferroni-type inequalities for P(& N (A) u, & M (B) v), where u and v are positive integers, are valid if and only if they are valid for a two dimensional triangular