Partitions of Integers - J. T. Butler Partitions of Integers

Example:

Definition:

There are five partitions of 4; 1+1+1+1, 2+1+1, 2+2, 3+1, and 4.

A partition of integer n is an integer solution to n = n1 + n2 + … + nk , where ni > 0 and n > 0. Specifically, this is a partition of n into k parts. Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

Note:

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A compact way to represent a partition is with exponents. That is, the partitions of 4 are 14, 212, 22, 31, and 4.

A partition of n is a way to distribute n nondistinct objects into n nondistinct cells with empty cells allowed. The order of the parts is not important. That is, 3+1 is the same as 1+3.

Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

If ms > 0, but ms+1 = ms+2 = … = mk = 0, then we have a partition of n −k into s parts. Each partition of n −k into s parts corresponds to a k part partition of n and vice versa.

Recurrence Relation for Partitions We can write a partition like so n = n1 + n2 +K+ n k ,

n1 ≥ n2 ≥ n3 ≥K ≥ n k ≥ 1

Now form n − k = ( n1 − 1) + ( n2 − 1) +K+ ( nk − 1) = m1 + m2 +K+ mk m1 ≥ m2 ≥K ≥ mk ≥ 0

Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler Thus, we can form the following table.

Thus, p k ( n ) = p k ( n − k ) + p k −1 ( n − k ) +K+ p 0 ( n − k ) ,

n k

where pk (n) is the number of partitions of n into k parts.

0 1 2 3 4 5 6 Sum

Example: p2 (4) = p2(2) + p1(2) + p0(2) 2 = 1 + 1 + 0 p4 (4) = p4(0) + p3(0) + p2(0) + p1(0) + p0(0) 1 = 0 + 0 + 0 + 0 + 1 Partitions of Integers - J. T. Butler

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In

k

(n)

k =1

Partitions of Integers - J. T. Butler

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1 + x 2 + x 4 + x 6 +K+ x 2 r +K =

4 0 1 2 1 1 0 0 5

5 0 1 2 2 1 1 0 7

6 0 1 3 3 2 1 1 11 ← p (n)

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1 + x + x 2 + x 3 +K+ x r +K =

1 1− x

the coefficient of xr is the number of ways to have r 1’s in a partition integer n.

Partitions of Integers - J. T. Butler

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If P (x) is the generating function for the number of partitions on n, then

Similarly, the polynomial

1 1− x2

P ( x ) = (1 + x + x 2 +K+ x r +K )(1 + x 2 + x 4 +K+ x 2 r +K ) (1 + x 3 + x 6 +K+ x 3r +K )(1 + x 4 + x 8 +K+ x 4 r +K )K 1 1 1 1 = K 1− x 1− x2 1− x3 1− x4 = 1 + x + 2 x 2 + 3x 3 + 5x 4 + 7 x 5 + 11x 6 +K

enumerates ways 2’s can appear in a partition of n.

Partitions of Integers - J. T. Butler

pk(n) 2 3 0 0 1 1 1 1 0 1 0 0 0 0 0 0 2 3

Generating Functions for Partitions

n

∑p

1 0 1 0 0 0 0 0 1

Partitions of Integers - J. T. Butler

The last line of the previous slide shows the number of partitions on n; that is,

p(n) =

0 1 0 0 0 0 0 0 1

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Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler Note:

Partitions with every part odd

It is convenient to represent special partitions using generating functions. Partitions with no repeated (summands are distinct)

S ( x) =

parts

1 1 1 K 3 1 − x 1 − x 1 − x5

R ( x ) = (1 + x )(1 + x 2 )(1 + x 3 )K Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

Note:

Example:

1 − x 2 1 − x 4 1 − x 6 1 − x 8 1 − x 10 K 1 − x 1 − x 2 1 − x 3 1 − x 4 1 − x5 1 1 1 = K 3 1 − x 1 − x 1 − x5

R( x ) =

Among the partitions on 4, there are two with all parts odd (31 and 14), and there are two with no repeated parts (31 and 4).

Thus, Theorem: The number of partitions with every part odd is the same as the number of partitions with no repeated parts. Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

Let

1 1 1 1 K = ∑ p≤ k ( n ) x n 1 − x 1 − x 2 1 − x 3 1 − x k n =0

x track the value of n y track the number of parts

n

where, p≤ k ( n) = ∑ pi ( n).

P( x, y ) = (1 + xy + x 2 y 2 + K)(1 + x 2 y + x 4 y 2 + K) K 1 1 1 = K (1) 2 1 − xy 1 − x y 1 − x 3 y

i=0

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Use generating functions to track the number of parts

Partitions with no part greater than k. Q( x) =

14

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Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler A typical term in this expression is

Example

pk(n) xn yk,

From the table for pk(n), the “coefficient” of

where pk(n) is the number of partitions of n into k parts.

y2 is [ 1 x2 + 1 x3 + 2 x4 + 2 x5 + 3 x6 + ... ] y2 12 12 13 14 15 22 23 24 32

A typical “coefficient” of yk in this expression is p ( k ) x k + p ( k + 1) x k +1 +K y k 1k44444k2444443

Pk ( x ) Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

Pk ( x ) y k − xy pk −1 ( k − 1) x k −1 + pk −1 ( k ) x k +K y k −1 =

Note:

pk ( k ) x k + pk ( k + 1) x k +1 +K x k y k

From (1), we can write (1 −xy) P (x,y) = P (x,xy)

Pk ( x ) y k − x Pk −1 ( x ) y k = x k Pk ( x ) y k (1 − x k ) Pk ( x ) = x Pk −1 ( x )

(2)

x Pk −1 ( x ) 1− xk x Pk −1 ( x ) = Pk −2 ( x ) 1 − x k −1

Equating the coefficient of yk on both sides of (2) yields

Partitions of Integers - J. T. Butler

and Pk ( x ) =

Pk ( x ) =

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Partitions of Integers - J. T. Butler

and

x2 Pk − 2 ( x ) (1 − x k )(1 − x k −1 )

Pk ( x ) =

Similarly,

Pk ( x ) =

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xk P0 ( x) (1 − x k )(1 − x k −1 ) K (1 − x ) 123 1

Note:

x3 Pk −3 ( x ) (1 − x k )(1 − x k −1 )(1 − x k −2 )

Pk ( x ) = x k Q ( x ), where Q ( x ) is the generating function for partitions with no part greater than k .

Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler Theorem:

Example:

The number of partitions of n with no part greater than k equals the number of partitions of n+k with exactly k parts.

There are four partitions of 6 with no part greater than 2 (16, 214, 2212, and 23). There are four partitions of 8 with exactly two parts (42, 53, 62, and 71).

Partitions of Integers - J. T. Butler

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Partitions of Integers - J. T. Butler

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that a represent the parts. This shows

Theorem: The number of partitions of n with the largest part k equals the number of partitions of n with exactly k parts.

6+4+3+1

Proof: Consider the Ferrer’s graph

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Partitions of Integers - J. T. Butler

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4+3+3+2+1+1

that a partition of 14 into 4 parts corresponds to a partition of 14 where the largest part is 4. The theorem statement follows.

of a partition, as shown below. Here, dots represent the parts. This shows Partitions of Integers - J. T. Butler

=

Partitions of Integers - J. T. Butler

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