# Recent questions tagged combinatory

1
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.
2
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than $n!$ ... respect to the correspondence between Cantor expansions and permutations as described in the preamble to question $14.$ $3$ $89$ $111$
3
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than $n!$ ... an algorithm for producing all permutations of a set of n elements based on the correspondence described in the preamble to question $14.$
4
Show that the correspondence described in the preamble is a bijection between the set of permutations of $\{1, 2, 3,\dots,n\}$ and the nonnegative integers less than $n!.$
5
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than $n!$ ... . Find the Cantor digits $a_{1}, a_{2},\dots,a_{n−1}$ that correspond to these permutations. $246531$ $12345$ $654321$
List all $3$-permutations of $\{1, 2, 3, 4, 5\}.$
Develop an algorithm for generating the $r$-permutations of a set of $n$ elements.
Show that Algorithm $3$ produces the next larger $r$-combination in lexicographic order after a given $r$-combination.