Date: Oct 16, 2007 3:08 AM
Author: China -->hk
Subject: who can help me?friends.
question 1: there are n numbers:1,2,...,n^2.now arrange them in a nxn matrix such that each row and each is increasing.Then what is the permutation number?

As we know

(from http://www.research.att.com/~njas/sequences/?q=matrix+increasing+in+rows+and+columns&language=english)

the formula is

T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! )

But can some respectful friend tell me how can we using high school knowledge(such as recurrence relation etc.,not using group)to deduce it out?If could,we can teach it in high school class.

question 2: if there an^3 numbers: 1,2,3,...,n^2,...,n^3,now arrange them in a cube (space) such that each row(x-axis),each column(y-axis) and each vertical line(z-axis) are strictly increasing,then what is the formula?

Thanks advancely,