partition identities related to stanley's...

Partition Identities Related to Stanley’s Theorem

Maxie D. Schmidt

Georgia Institute of Technology

maxieds@gmail.com

mschmidt34@gatech.edu

October 9, 2017

Maxie D. Schmidt (GA Tech) Partition Identities October 9, 2017 1 / 31

Overview

1 Agenda

2 Partitions and Special Partition FunctionsEuler’s Partition FunctionOther Special Partition FunctionsRestricted Partition Functions

3 Stanley’s TheoremStatement of the Theorem and ExamplesAnother Form of Stanley’s TheoremProof of the Theorem via Generating Functions

4 New Formulas for Euler’s Partition Function

5 Concluding Remarks

Agenda

Agenda Today

Discuss the definition of partitions and various partition functions

Connect restricted partitions with the functions φ(n) and µ(n) frommultiplicative number theory

Get a working understanding of Stanley’s theorem and its variants

Prove new identities for Euler’s partition function p(n) involvingunusual connections to special multiplicative functions

Agenda

Why are these new results important?

So why are these identities special?These results are significant and special in form given their unusualnew connections with the functions of multiplicative number theoryand the more additive nature of the theory of partitions.

Formulas and identities connecting the at times seemingly disparatebranches of additive and multiplicative number theory are rare, withonly a few good examples of these types known in the literature.

This talk is primarily based on joint work with Mircea Merca acceptedto appear in the American Mathematical Monthly in 2018.

Partitions and Special Partition Functions Euler’s Partition Function

The Partition Function p(n)

Partitions of n (Euler’s Partition Function)

A partition of a positive integer n is a sequence of positive integerswhose sum is n

More formally, we may write a partition of n as a sum of positiveintegers λ1 ≥ λ2 ≥ · · · ≥ λk ≥ 1 such that λ1 + λ2 + · · ·+ λk = n

We denote the total number of partitions of n by p(n)

For example, p(5) = 7 since the partitions of 5 are given as:

5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1.

Partitions of n (Generating Function)

By a combinatorial argument, the generating function for p(n) is∑n≥0

p(n)qn =1

(q; q)∞=

(1− q)(1− q2)(1− q3) · · ·,

where (a; q)n = (1− a)(1− aq) · · · (1− aqn−1) denotes theq-Pochhammer symbol

Why (you may ask)?

Partitions of n (Generating Function)

By the representation of the partitions of 5 given above, we may writethe number of partitions of n as the number of positive integersolutions to the equation x1 + 2x2 + 3x3 + · · ·+ nxn = n

Moreover, by expanding geometric series we can count by expandingthe generating function products and taking the coefficient of qn∑n≥0

p(n)qn =1

(1− q)(1− q2)(1− q3) · · ·

=∞∏k=1

(1 + qk + q2k + q3k + · · ·

)= (1 + · · ·+ qx1 + · · · )

(1 + · · ·+ q2·x2 + · · ·

) (1 + · · ·+ q3·x3 + · · ·

)· · ·

Other Special Partition Functions

Partitions and Special Partition Functions Other Special Partition Functions

Other Special Partition Functions

q(n) := [qn](q; q2)−1∞ is another special common partition functionwhich denotes the number of partitions of n into distinct parts, orequivalently into odd parts (see A000009)Other special partition functions are commonly defined by theirq-series generating functions as in the following table

Generating Function The number of partitions of n into parts which are

∞∏m=1

11−q2m−1 odd

∞∏m=1

11−q2m even

∞∏m=1

1−qm2 squares

Table: Partition Functions by Their Generating Functions

Partitions and Special Partition Functions Other Special Partition Functions

Other Special Partition Functions (cont.)

Generating Function The number of partitions of n into parts which are∏p prime

11−qp primes

∞∏m=1

(1 + qm) unequal

∞∏m=1

(1 + q2m−1) odd and unequal

∞∏m=1

(1 + q2m) even and unequal

∞∏m=1

(1 + qm2) distinct squares∏

p prime(1 + qp) distinct primes

Table: Partition Functions by Their Generating Functions

Partitions and Special Partition Functions Restricted Partition Functions

Restricted Partition Functions

Applying the counting argument above by generating functions, wecan generate the number of partitions of n with parts at least r by(1− q)(1− q2) · · · (1− qr−1)/(q; q)∞

Similarly we may define the sequence S(r)n,k for n, k , r ≥ 1 to be the

number of k’s in the partitions of n with smallest part at least r

The sequence of restricted partition functions, S(r)n,k , is primarily

motivated by its combinatorial interpretation of the generatingfunction (as we shall see):

∑n≥0

S(r)n,kq

n =(1− q)(1− q2) · · · (1− qr−1)

(q; q)∞· qk

1− qk

Stanley’s Theorem

Stanley’s Theorem Statement of the Theorem and Examples

Stanley’s Theorem

Theorem (Stanley)

The number of 1’s in the partitions of n is equal to the number of partsthat appear at least once in a given partition of n, summed over allpartitions of n.

Stanley’s Theorem Statement of the Theorem and Examples

An Example of Stanley’s Theorem

Example (MathWorld)

We check the theorem when n = 5 by an example.

Written in set notation, the partitions of 5 are given as follows:{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}.Then there are a total of 0 + 1 + 0 + 2 + 1 + 3 + 5 = 12 distinct 1’s inall of the partitions of n, and moreover,

The number of unique terms in each partition of 5 is given by1 + 2 + 2 + 2 + 2 + 2 + 1 = 12.

Thus the theorem holds when n = 5 (see A000070).

Proof of Stanley’s Theorem

We do not give an explicit proof of Stanley’s theorem here. However, wedo prove the next restatement of it in terms of restricted partitionfunctions.

Stanley’s Theorem Another Form of Stanley’s Theorem

Another Form of Stanley’s Theorem

Euler’s Totient Function φ(n)

Definition and Generating Functions

Euler’s totient function, φ(n), is defined for all n ≥ 1 as

φ(n) =∑

d :(d,n)=1

1 7→ {1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, . . . }

(see sequence A000010).Euler’s totient function is multiplicative, i.e., φ(mn) = φ(m)φ(n) whenever(m, n) = 1, and since n =

∑d|n φ(d) for all n ≥ 1, it has a “nice” Lambert series

generating function given by∑n≥1

φ(n)qn

1− qn=

(1− q)2.

A Restatement of Stanley’s Theorem

Theorem

The number of 1’s in the partitions of n is equal to

S(1)n,1 =

n+1∑k=2

φ(k)S(2)n+1,k ,

where the restricted partition function S(2)n,k counting the number of ks in

the partitions of n with smallest part greater than 1 is generated by

S(2)n,k = [qn]

(1− q)

(q; q)∞· qk

1− qk

b nk c∑i=1

(p(n − i · k)− p(n − 1− i · k)) .

A Table of the S(2)n,k (n, k ≥ 2)

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 04 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 03 2 1 1 0 1 0 0 0 0 0 0 0 0 0 0 08 3 3 1 1 0 1 0 0 0 0 0 0 0 0 0 07 6 2 2 1 1 0 1 0 0 0 0 0 0 0 0 0

15 6 5 3 2 1 1 0 1 0 0 0 0 0 0 0 015 10 5 4 2 2 1 1 0 1 0 0 0 0 0 0 027 14 10 5 5 2 2 1 1 0 1 0 0 0 0 0 029 18 10 8 4 4 2 2 1 1 0 1 0 0 0 0 048 24 17 10 8 5 4 2 2 1 1 0 1 0 0 0 0

A Table of the S(2)n,k Inverse Matrix (n, k ≥ 2)

1 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0−2 0 1 0 0 0 0 0 0 0 0 0 0 0−1 −1 0 1 0 0 0 0 0 0 0 0 0 0−2 −2 −1 0 1 0 0 0 0 0 0 0 0 00 −1 −1 −1 0 1 0 0 0 0 0 0 0 01 0 −2 −1 −1 0 1 0 0 0 0 0 0 01 −1 0 −1 −1 −1 0 1 0 0 0 0 0 01 2 0 −1 −1 −1 −1 0 1 0 0 0 0 01 1 1 0 0 −1 −1 −1 0 1 0 0 0 04 2 1 1 −1 0 −1 −1 −1 0 1 0 0 01 1 1 1 1 0 0 −1 −1 −1 0 1 0 0−1 2 2 2 1 0 0 0 −1 −1 −1 0 1 01 0 1 0 1 1 1 0 0 −1 −1 −1 0 1

Stanley’s Theorem Proof of the Theorem via Generating Functions

Proof of Theorem Theorem 1

We can expand the series

1− q

(q; q)∞

∞∑k=1

φ(k)qk

1− qk=

(q; q)∞+∞∑n=2

∞∑k=2

φ(k)S(2)n,kq

On the other hand, considering the well-known Lambert series generatingfunction [HardyWright, Theorem 309]

∞∑k=1

φ(k)qk

1− qk=

(1− q)2,

we obtain

1− q

(q; q)∞

∞∑k=1

φ(k)qk

1− qk=

(q; q)∞+

1− q· q

(q; q)∞

(q; q)∞+∞∑n=1

S(1)n,1q

Stanley’s Theorem Proof of the Theorem via Generating Functions

Proof of Theorem Theorem 1 (cont’d)

Thus, we deduce the identity

S(1)n,1 =

n+1∑k=2

φ(k)S(2)n+1,k ,

for any n ≥ 1, which completes the proof of the theorem.

New Formulas for Euler’s Partition Function

New Formulas for p(n)

A New Formula for the Partition Function

Theorem

For n ≥ 0,

p(n) =n+3∑k=3

Pφ(k)S(3)n+3,k ,

where Pφ(k) := φ(k)/2.

The Mobius Function µ(n)

Definition

The Mobius function, µ(n), is the multiplicative arithmetic functiondefined by

µ(n) =

1, if n = 1;

(−1)ν(n), if a1 = a2 = · · · = aν(n) = 1;

0, otherwise,

when the factorization of n into distinct prime factors is given byn = pa11 pa22 · · · p

aν(n)ν(n) and where ν(n) denotes the number of distinct prime

factors dividing n.The Lambert series generating function for µ(n) is given by [HardyWright,Theorem 308] ∑

µ(n)qn

1− qn= q.

A Related Formula for p(n)

Corollary (Expansion by the Mobius Function)

For all n ≥ 1, we have the identity that

p(n) = −n∑

j∑k=0

µ(k + 3)S(3)j+3,k+3.

Wrapping Up: Other Generalized Formulas for p(n)Involving the Mobius Function

Since for α ≥ 2 prime we can expand the Lambert series for µ(αn) as∑n≥1

µ(αn)qn

1− qn= −

∑n≥0

we similarly can prove the following two special case identities providingeven further new recurrence relations for p(n):

n+1∑k=1

S(1)n+1,kµ(αk) = −

∑k>0

p(n + 1− αk)

n+2∑k=2

S(2)n+2,kµ(αk) =

∑k>0

(p(n + 1− αk)− p(n + 2− αk+1)

Concluding Remarks

Summary

We have connected Euler’s totient function with a new decompositionof Stanley’s theorem.

Along the way, we have derived a new formula for Euler’s partitionfunction p(n) expanded in terms of our additive restricted partition

functions S(r)n,k when r = 3 and duplicated the proof of this result to

obtain new analogous formulations for p(n) involving the Mobiusfunction.

We have defined and computed several generalized forms of bothidentities for p(n) involving the restricted partition functions and φ(n)and µ(n), respectively.

Concluding Remarks

References

G. E. Andrews, The Theory of Partitions. Addison-Wesley Publishing, New York,1976.

T. M. Apostol, Introduction to Analytic Number Theory. Springer, New York, 1976.

G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers. ClarendonPress, Oxford, 1979.

M. Merca and M. D. Schmidt, A Partition Identity Related to Stanley’s Theorem,Amer. Math. Monthly, to appear in 2018.

M. Merca and M. D. Schmidt, Euler’s Partition Function in Terms of the ClassicalMobius Function, submitted to Ramanujan J. as of September 2017.

Concluding Remarks

The End.Thank you for listening.Questions or comments?

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