CO40 Generating Functions: Combinations of Multisets

How many non-negative integer solutions to x_1+x_2+x_3= 47 if x_1 is a multiple of 7, 0 ≤ x_2 ≤ 6, and x_3 ≥ 0? Using ordinary generating functions to find the number of combinations of a multiset. Equivalently, find the number of non-negative integer solutions to x_1 + ... + x_t = s when there are different kinds of conditions on each of x_1, ..., x_t. Examples and Proofs. Examples include finding p(11) the number of integer partitions of 11 using Taylor polynomials. Material organized with the use of indicator power series. Subscribe @Shahriari for more undergraduate math videos.
00:00 Introduction
00:27 Guiding Problem: Finding the number of combinations of a multiset with unusual restrictions
01:21 Equivalent formulations of the guiding problem
03:37 Example: Find the number of non-negative integer solutions to x_1+x_2+x_3=47 if x_1 is a multiple of 7, 0 ≤ x_2 ≤ 6, and x_3 ≥ 0
08:33 Indicator Power Series
09:43 Example: Indicator Power Series for the set of positive even integers
11:04 Example: Indicator Power Series for 0, 1, ..., 10
20:59 The main result on counting combinations of multisets and its many interpretations
24:45 Example: the solution to the "Guiding Problem"
29:15 Example: the number of non-negative integer solutions to x_1+x_2+x_3=47 if x_1 is a multiple of 7, 0 ≤ x_2 ≤ 6, and x_3 ≥ 0
32:05 Definition of Partition Numbers p(n)
33:16 Instead of finding p(n) find the number of certain non-negative integer solutions
35:20 Finding p(11) using ordinary generating functions

A series of lectures on introductory Combinatorics. This full course is based on my book
Shahriar Shahriari, An Invitation to Combinatorics, Cambridge University Press, 2022.

For an annotated list of available videos see

Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont CA, USA
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