scholarly journals Convolution property and exponential bounds for symmetric monotone densities

2013 ◽  
Vol 17 ◽  
pp. 605-613 ◽  
Author(s):  
Claude Lefèvre ◽  
Sergey Utev
Keyword(s):  
Exponential Bounds ◽  
Monotone Densities

scholarly journals On the Number of Distributive Lattices

10.37236/1641 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Marcel Erné ◽  
Jobst Heitzig ◽  
Jürgen Reinhold
Keyword(s):  
Combinatorial Identities ◽  
Distributive Lattices ◽  
Integer Sequences ◽  
Isomorphism Classes ◽  
Exponential Bounds

We investigate the numbers $d_k$ of all (isomorphism classes of) distributive lattices with $k$ elements, or, equivalently, of (unlabeled) posets with $k$ antichains. Closely related and useful for combinatorial identities and inequalities are the numbers $v_k$ of vertically indecomposable distributive lattices of size $k$. We present the explicit values of the numbers $d_k$ and $v_k$ for $k < 50$ and prove the following exponential bounds: $$ 1.67^k < v_k < 2.33^k\;\;\; {\rm and}\;\;\; 1.84^k < d_k < 2.39^k\;(k\ge k_0).$$ Important tools are (i) an algorithm coding all unlabeled distributive lattices of height $n$ and size $k$ by certain integer sequences $0=z_1\le\cdots\le z_n\le k-2$, and (ii) a "canonical 2-decomposition" of ordinally indecomposable posets into "2-indecomposable" canonical summands.

Explicit and exponential bounds for a test on the coefficient of an AR(1) model

1995 ◽  
Vol 25 (4) ◽  
pp. 365-371 ◽  
Author(s):  
Bruno Massé ◽  
Marie-Claude Viano
Keyword(s):  
Exponential Bounds
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