More about partitions

Transcription

1 Partitions 2.4, 3.4, More about partitions 3 + +, + 3 +, and are all the same partition, so we will write the numbers in non-increasing order. We use greek letters to denote partitions, often λ ( lambda ), µ ( mu ), and ν ( nu ). We ll write: λ : n = n + n n k or λ n. For example, λ : 5 = 3 + +, or λ = 3, or λ =3 2, or 3 5. A pictoral representation of λ = n n 2 n k is its Ferrers diagram, a left-justified array of dots with k rows, containing n i dots in row i. Example. The Ferrers diagram of is The conjugate of a partition λ is the partition λ c which interchanges rows and columns. Some partitions are self-conjugate, satisfying λ = λ c.

2 Partitions 2.4, 3.4, A generating function for partitions Recall from our basketball example: The generating function for the number of ways to partition an integer into parts of size, 2, or 3 is ( x) ( x 2 ) ( x 3 ) If we include parts of any size, we infer: Let p(n) be the number of partitions of the integer n. Then p(n)x n = x k n 0 k= Notes: Infinite product! But, for any n only finitely many terms involved. There is a beautiful generating function, but no nice formula! Finding a generating function for a subset of partitions is easy if you understand each factor in the product.

3 Partitions 2.4, 3.4, A formula for integer partitions Weisstein, Eric W. Partition Function P. From MathWorld A Wolfram Web Resource.

4 Partitions 2.4, 3.4, Partitions: odd parts and distinct parts Example. THE FOLLOWING AMAZING FACT!!!!!!!!! The number of partitions of n using only odd parts, o n = Investigation: Does this make sense? For n = 6, o 6 : d 6 : Solution. Determine the generating functions The number of partitions of n using distinct parts, d n O(x) = n 0 o nx n D(x) = n 0 d nx n See, I told you they were equal.

5 Partitions 2.4, 3.4, A recurrence relation for P(n, k) (p.78) Example. Prove a recurrence relation for P(n, k): P(n, k) = P(n, k ) + P(n k, k) Question: How many partitions of n are there into k parts? LHS: P(n, k) RHS: Condition on whether the smallest part is of size. If so, there are P(n, k ) partitions via the bijection { } { partitions of n into k parts partitions of n f : with smallest part. into k parts. If not: there are P(n k, k) partitions via the bijection { } { partitions of n into k parts partitions of n k g : with smallest part. into k parts. }. }.

6 Partitions 2.4, 3.4, Using conjugation Theorem P(n, k) equals P(n, largest part = k) Proof. The conjugation function f : λ λ c is a bijection { } { partitions of n partitions of n with f : into exactly k parts largest part of size k. }. The same bijection gives: Theorem equals P(n, largest part k).

7 Partitions 2.4, 3.4, Characterization of self-conjugate partitions Theorem P(n, self conjugate) =P(n, distinct odd parts) Proof. Define a bijection which unfolds self-conjugate partitions: { } { } self-conjugate partitions µ of n into f :. partitions λ of n distinct odd parts Define parts of µ by unpeeling λ layer by layer. Iteratively remove the first row and first column of λ. Question: Is f well defined? Define the inverse function g = f : µ λ: Find the center dot of each part µ i. Fold each µ i about its center dot. Nest these folded parts to create λ. Question: Is g well defined? Question: Is g(f (λ)) = λ?

8 Formulas for P(n, 2) and P(n, 3) Partition Formulas A formula for P(n) is hard. We can find formulas for P(n, 2) & P(n, 3). Theorem. P(n, 2) =. Proof. Theorem P(n, 3) is the closest integer to n 2 /2. Proof. Above theorems give us: P(n, 3) = P(n 3, at most 3 parts) = P(n 3, parts of size 3). Question: How many partitions of n 3 into parts of size, 2, 3? Answer: The generating function is ( x)( x 2 )( x 3 ) = /6 + /4 + /3 + /4. ( x) 3 ( x) 2 x 3 x 2 The coefficient of x n 3 is 6 ( n ) Simplifying, 6 ( n 2 (( 3 n 3 )) + 4 ) = n (( 2 )) n 3 +[ 3 or 0] + [ 4 or 0].

9 Advanced Topic: Standard Young Tableaux 0 Standard Young Tableaux Related to some current lines of research in algebra and combinatorics: A Young diagram is a representation of a partition using left-justified boxes. A standard Young tableau is a placement of the integers through n into the boxes, where the numbers in both the rows and the columns are increasing The hook length h(i, j) of a cell (i, j) is the number of cells in the hook to the left and down. 5 Question: How many SYT are there of shape λ n? Answer: (i,j) λ n! h(i, j)