Abstract
A preferential arrangement on [ [ n ] ] = { 1 , 2 , … , n } is a ranking of the elements of [ [ n ] ] where ties are allowed. The number of preferential arrangements on [ [ n ] ] is denoted by r n . The Delannoy number D ( m , n ) is the number of lattice paths from ( 0 , 0 ) to ( m , n ) in which only east ( 1 , 0 ) , north ( 0 , 1 ) , and northeast ( 1 , 1 ) steps are allowed. We establish a symmetric identity among the numbers r n and D ( p , q ) by means of algebraic and combinatorial methods.
Highlights
A preferential arrangement on [[n]] = {1, 2, . . . , n} is a ranking of the elements of [[n]] where ties are allowed
We use the result in Lemma 1 to get some relations between Delannoy numbers and preferential arrangements
The definition of the stuffle product ∗ indicates that the stuffle product of two multiple zeta values of depth m and n will produce D(m, n) numbers of multiple zeta values ([25])
Summary
As a preferential arrangement is nothing else than a sequence of non-empty sets, this directly gives the generating function SEQ(SET≥1 (Z ))= 1/(2 − exp(z)), and this explains the recurrence and the link with Stirling numbers The explicit expression of D (m, n) and the generating function [1] are given by: m m m+n−k The significances of these numbers are explained in [10,11]. We introduce combinatorial viewpoints to approach the Main Theorem in the last section
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