Integer Partitions
A. Satyanarayana Reddy
Department of Mathematics
School of Natural Sciences
Shiv Nadar Institution of Eminence, Delhi-NCR
October 8, 2024
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Integer Partitions: Concepts, Properties, and Bijections, Lecture notes of Combinatorics
This document delves into the fascinating world of integer partitions, exploring their fundamental definitions, properties, and relationships. It introduces the concept of parts in a partition and provides examples to illustrate the different ways to partition a positive integer. The document also explores the concept of conjugate partitions and how they can be used to prove various properties of partitions. It presents three problems related to partitions and encourages the reader to prove them using conjugate partitions and bijections. The document further explores the concept of partitions with distinct parts and even distinct parts, providing examples and demonstrating their properties. It concludes with a discussion of partitions into prime parts and square parts, highlighting their significance in number theory.
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2023/2024
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Integer Partitions
A. Satyanarayana Reddy
Department of Mathematics
School of Natural Sciences
Shiv Nadar Institution of Eminence, Delhi-NCR
October 8, 2024
Outline
Introduction
Partitions with restrictions
Few Problems
Euler Pentagonal Theorem
- 1 =
- 1 =
- 2 = 1 +
- 1 =
- 2 = 1 +
- =
- 1 =
- 2 = 1 +
- =
- 3 = 1 + 1 +
- 1 =
- 2 = 1 +
- =
- 3 = 1 + 1 +
- = 2 +
- 1 =
- 2 = 1 +
- =
- 3 = 1 + 1 +
- = 2 +
- =
- 1 =
- 2 = 1 +
- =
- 3 = 1 + 1 +
- = 2 +
- =
- 4 = 1 + 1 + 1 +
- = 2 + 1 +
- = 2 +
- = 3 +
- =
- 1 =
- 2 = 1 +
- =
- 3 = 1 + 1 +
- = 2 +
- =
- 4 = 1 + 1 + 1 +
- = 2 + 1 +
- = 2 +
- = 3 +
- =
- 5 = 1 + 1 + 1 + 1 +
- = 2 + 1 + 1 +
- = 2 + 2 +
- = 3 + 1 +
- = 3 +
- = 4 +
- =
- 1 =
- 2 = 1 +
- =
- 3 = 1 + 1 +
- = 2 +
- =
- 4 = 1 + 1 + 1 +
- = 2 + 1 +
- = 2 +
- = 3 +
- =
- 5 = 1 + 1 + 1 + 1 +
- = 2 + 1 + 1 +
- = 2 + 2 +
- = 3 + 1 +
- = 3 +
- = 4 +
- =
- 6 = 1 + 1 + 1 + 1 + 1 +
- = 2 + 1 + 1 + 1 +
- = 2 + 2 + 1 +
- = 3 + 1 + 1 +
- = 2 + 2 +
- = 3 + 2 +
- = 4 + 1 +
- = 3 +
- = 4 +
- = 5 +
- =
Definition
A partition of a positive integer n is defined to be sequence of postive
integers whose sum is n. The order of the summands is unimportant when
writing the partion of integer, but summands should be non-increasing
order.
Definition
A partition of a positive integer n is defined to be sequence of postive
integers whose sum is n. The order of the summands is unimportant when
writing the partion of integer, but summands should be non-increasing
order.
Given n ∈ N a k-tuple λ = ( λ 1
, λ 2 ,... , λ k
) ∈ N
k is a partition of n if
▶ (^) n = λ 1
- λ 2
- · · · + λ k
▶ (^) λ 1 ≥ λ 2 ≥ · · · ≥ λ k
≥ 1
We write λ ⊢ n.
(2, 1) ⊢ 3, ( 3 ) ⊢ 3, (1, 1, 1) ⊢ 3
▶ (^) A summand in a partion is called part.
▶ (^) For example, 2, 1 are parts of the partition 3 = 2 + 1.
▶ (^) A summand in a partion is called part.
▶ (^) For example, 2, 1 are parts of the partition 3 = 2 + 1.
▶ (^) Let p(n) counts the number of parttions of a postive integer n. We take
p(n) = 0 for all negative values of n and p( 0 ) is defined to be 1.
▶ (^) A summand in a partion is called part.
▶ (^) For example, 2, 1 are parts of the partition 3 = 2 + 1.
▶ (^) Let p(n) counts the number of parttions of a postive integer n. We take
p(n) = 0 for all negative values of n and p( 0 ) is defined to be 1.
▶ Then p( 0 ) = 1, p( 1 ) = 1, p( 2 ) = 2, p( 3 ) = 3, p( 4 ) = 5, p( 5 ) = 7, p( 6 ) =
11, p( 7 ) = 15, p( 8 ) = 22, p( 9 ) = 30, p( 10 ) = 42.
▶ What is the value of p( 11 )?.
▶ (^) A summand in a partion is called part.
▶ (^) For example, 2, 1 are parts of the partition 3 = 2 + 1.
▶ (^) Let p(n) counts the number of parttions of a postive integer n. We take
p(n) = 0 for all negative values of n and p( 0 ) is defined to be 1.
▶ Then p( 0 ) = 1, p( 1 ) = 1, p( 2 ) = 2, p( 3 ) = 3, p( 4 ) = 5, p( 5 ) = 7, p( 6 ) =
11, p( 7 ) = 15, p( 8 ) = 22, p( 9 ) = 30, p( 10 ) = 42.
▶ What is the value of p( 11 )?.
▶ (^) Is p(n) is increasing?