Partitions functions

Created on March 15, 2025

2025

Here’s a presentation I did in my master’s you can go through it : link or read this blog for glimpses.

Partitions

A partition of a positive integer \(n\) is a finite nonincreasing sequence of positive integers : \((\lambda_1,\lambda_2,\dots,\lambda_r) \text{ such that } \sum_{i=1}^r \lambda_i = n.\) where \(\lambda_i\)’s are called the parts of the partition

The partition function \(p(n)\) is the number of partitions of \(n\) for example: \(\begin{align*} p(1) &= 1: 1=(1)\\ p(2) &= 2: 2 = (2), 1 + 1 = (1^2 )\\ p(3) &= 3: 3 = (3), 2 + 1 =(2\; 1), 1 + 1 + 1 = (1^3)\\ p(4) &= 5: 4 = (4), 3 + 1 =(3\; 1), 2 + 2 = (2^2),2 + 1 + 1 = (2^2\; 1), 1 + 1 + 1 + 1 = (1^4). \end{align*}\)

The partition function \(p(n)\) increases quite rapidly with \(n\). For example, \(p(10) = 42, p(20) = 627, p(50) = 204226, p(100) = 190569292, p(200) =3972999029388.\)

Basic q series representations

  • we define \((a;q)_n=(1-a)(1-aq)\dots (1-aq^{n-1})\) this it is a type of Pochhammer notation of shifted factorial extended to base ‘\(q\)’
  • also \((q;q)_n=(1-a)(1-aq)\dots (1-q^{n})\) which is a very tipical notation
  • if \(f_k=(q^k;q^k)_\infty=(1-q^k)(1-q^{2k})\dots(1-q^{nk})\dots\) now from partition theory we have :
  • if \(p(n)\) represent the number of integer partitions of a non negative integer n then :
\[\begin{align*} \sum_{n=0}^\infty p(n)q^n &= \frac{1}{(q;q)_\infty}=\frac{1}{f_1}\\ \qquad &=(1-q)^{-1}(1-q^2)^{-1}\dots(1-q^n)^{-1}\dots \\ \qquad &= (1+q+q^2+q^3+\dots+q^n+\dots)(1+q^2+q^4+q^6+\dots+q^2n+\dots)\\ &\quad \dots(1+q^n+q^{2n}+q^{3n}+\dots)\dots \end{align*}\]

( these expansions holds for \(|q|<1\) )

  • when this last equation is restricted to say \(q^m\) terms and no more, we can then expand and find coefficients of the terms to get \(p(m)\) or other less terms \(<m\)

restricted partitions

  • a \(\; l-regular\) partition of n is partition of n in which no part is a multiple of \(l\) (a positive integer)
  • which in \(q\) series can be represented as:
  • if \(p_l(n)\) represents the number of integer partitions of a non negative integer n in which no part is divisible by \(l\) then we have :
\[\begin{align*} \sum_{n=0}^\infty p_l(n)q^n &= \frac{(q^l;q^l)_\infty}{(q;q)_\infty}=\frac{f_l}{f_1}\\ \qquad &=(1-q)^{-1}(1-q^2)^{-1}\dots(1-q^{l-1})^{-1}(1-q^{l+1})^{-1}\\ &\quad \dots (1-q^{2l-1})^{-1}(1-q^{2l+1})^{-1}\dots\\ \qquad &= (1+q+q^2+\dots)(1+q^2+q^4+\dots)\dots\\ &\quad (1+q^{l-1}+q^{2(l-1)}+\dots)(1+q^{l+1}+q^{2(l+1)}+\dots)\dots \end{align*}\]
  • a \((l,j)-regular\) partition of n is partition of n in which no part is a multiple of \(l\) or \(j\)
    (which are positive integer and co-prime i.e \(gcd(l,j)=1\))
  • which in \(q\) series can be represented as:
  • if \(p_{l,j}(n)\) represents the number of integer partitions of a non negative integer n in which no part is divisible by \(l\) or \(j\) , \(gcd(l,j)=1\) then we have
\[\begin{align*} \sum_{n=0}^\infty p_{l,j}(n)q^n &= \frac{(q^l;q^l)_\infty (q^j;q^j)_\infty}{(q;q)_\infty (q^{jk};q^{jk})_\infty} \qquad = \frac{f_l f_j}{f_1 f_{lj}}\\ \qquad &= (1-q)^{-1}(1-q^2)^{-1}\dots(1-q^{l-1})^{-1}(1-q^{l+1})^{-1}\dots (1-q^{j-1})^{-1}(1-q^{j+1})^{-1}\dots\\ \qquad &= (1+q+q^2+\dots)(1+q^2+q^4+\dots)\dots(1+q^{l-1}+q^{2(l-1)}+\dots)\\ &\quad(1+q^{l+1}+q^{2(l+1)}+\dots)\dots (1+q^{j-1}+q^{2(j-1)}+\dots)(1+q^{j+1}+q^{2(j+1)}+\dots)\dots \end{align*}\]

Colored paartitions

  • Now clearly to create a partition of \(n\) parts \((\lambda_i)\) are chosen from \(N\) (set of natural numbers = 1,2,3,…) let this set of parts to be chosen represented by \(S\) then:
    • for partition of \(n \quad S=N\)
    • for \(l-regular\) partition of \(n\quad S = N - l \; N = \{1,\dots ,l-1,l+1,\dots,2l-1,2l+1,\dots\)
    • for \((l,m)-regular\) partition of \(n \quad S = N - l \; N - m \; = \{1,\dots ,l-1,l+1,\dots,m-1,m+1,\dots\}\)
  • Now if \(S=N^s=\{1,1,\dots,2,2,\dots\}\) i.e. each part is repeated s times and each part is distinct from other
  • for our convenience and without loss of generality! we give each of this distinct part a color like \(1_r\) is 1 with red color, \(1_b\) is one with blue color etc
  • from above we have \(N^s={1_r,1_b,\dots,2_r,2_b,\dots}\) i.e.each part has \(s\) distinct colors and is unique.