(3) Recall that a lattice path on the grid from (0,0) to (n, n) can be viewed as a sequence of length 2n which contains n-many R's and n-many U's. Let C. be the number of such paths which never go above the diagonal, i.e., those sequences in which there are never more U's than R's in any initial segment. Show that C, is equal to the number of ways of placing n unlabelled balls inn labelled urns so that for each k with 1sksn there are at most k-many balls total in the first k-many urns.

(3) Recall that a lattice path on the grid from (0,0) to (n, n) can be viewed as a sequence of length 2n which contains n-many R's and n-many U's. Let C. be the number of such paths which never go above the diagonal, i.e., those sequences in which there are never more U's than R's in any initial segment. Show that C, is equal to the number of ways of placing n unlabelled balls inn labelled urns so that for each k with 1sksn there are at most k-many balls total in the first k-many urns.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 74EQ

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