Complementary Schur Asymptotics for Partitions

TL;DR

This paper derives asymptotic formulas for the number of partitions avoiding certain parts, extending classical results and constructing complex partition ideals.

Contribution

It introduces asymptotics for restricted partition functions based on Hardy-Ramanujan results, enabling the construction of oscillating partition ideals.

Findings

Derived asymptotics for partitions excluding finite sets of parts

Extended classical Hardy-Ramanujan asymptotics to restricted partitions

Constructed examples of highly oscillating partition ideals

Abstract

We deduce from the strong form of the Hardy--Ramanujan asymptotics for the partition function $p(n)$ an asymptotics for $p_{-S}(n)$, the number of partitions of $n$ that do not use parts from a finite set $S$ of positive integers. We apply this to construct highly oscillating partition ideals.

Equations

$$\displaystyle p(n)\sim\frac{e^{C\sqrt{n}}}{4n\sqrt{3}}\;\mbox{ where $C=\pi\sqrt{2/3}\approx 2.565$}\;.$$

$$p_S(n)\sim \frac{n^{t-1}}{(t-1)!{\prod }_{s\in S}s}=\frac{1}{n(t-1)!}\prod _{s\in S}\frac{n}{s}.$$

$$p_{-S}(n)\sim p(n){\left(\frac{C}{2\sqrt{n}}\right)}^t\prod _{s\in S}s=p(n)\prod _{s\in S}\frac{Cs}{2\sqrt{n}}.$$

$${\lambda }_n:=\sqrt{n-\frac{1}{24}},C:=\pi \sqrt{\frac{2}{3}}\text{and}D\in (C/2,C)$$

$$p(n)=\frac{e^{C{\lambda }_n}}{4\pi \sqrt{2}{\lambda }_n^2}\left(C-\frac{1}{{\lambda }_n}\right)+O\left(e^{D\sqrt{n}}\right).$$

$$p_{-S}(n)\sim p(n){\left(\frac{C}{2\sqrt{n}}\right)}^t\prod _{s\in S}=p(n)\prod _{s\in S}\frac{Cs}{2\sqrt{n}}.$$

$$q(n):=\frac{e^{C{\lambda }_n}}{{\lambda }_n^2}\left(C-\frac{1}{{\lambda }_n}\right).$$

$$p(n-s)=\frac{e^{C\sqrt{n}}}{4\pi n\sqrt{2}}\sum _{z=0}^{t}g(z,s)n^{-z/2}+O\left(e^{C\sqrt{n}}{n}^{-\frac{t+3}{2}}\right)$$

$$g(z,s)=\frac{(-1{)}^z{C}^{z+1}}{2^{z}z!}{s}^z+h(z,s)$$

$$e^{C{\lambda }_n}$$

$$\frac{1}{{\lambda }_n^2}$$

$$C-\frac{1}{{\lambda }_n}$$

$$q(n)=\frac{e^{C{\lambda }_n}}{{\lambda }_n^2}\left(C-\frac{1}{{\lambda }_n}\right)=e^{C\sqrt{n}}\left(\sum _{k=0}^t{a}_k{n}^{-k/2-1}+O(n^{-\frac{t+3}{2}})\right)$$

$$\sum _{k=0}^t{a}_k(n-s{)}^{-k/2-1}$$

$$=\sum _{k=0}^t{a}_k\left(\sum _{l=0}^t\left(\genfrac{}{}{0ex}{}{-1-k/2}{l}\right)(-s{)}^l{n}^{-1-k/2-l}+O(n^{-k/2-t-2})\right)$$

$$=\sum _{w=0}^{2t}f(w,s)n^{-w/2-1}+O(n^{-t-2})$$

$$f(w,s)=\sum _{k,l\ge 0,k+2l=w}(-1{)}^l{a}_k(\genfrac{}{}{0ex}{}{-1-k/2}{l})s^l$$

$$e^{C\sqrt{n-s}}$$

$$d(i,s)=\frac{(-1{)}^i{C}^i}{2^{i}i!}{s}^i+c_{i-1}{s}^{i-1}+\dots$$

$$q(n-s)=$$

$$=$$

$$g(z,s)=\sum _{i,w\ge 0,i+w=z}d(i,s)f(w,s)$$

$$g(z,s)=\frac{(-1{)}^z{C}^{z+1}}{2^{z}z!}{s}^z+h(z,s)$$

$$p(n-s)$$

$$=\frac{e^{C\sqrt{n}}}{4\pi n\sqrt{2}}\sum _{z=0}^{t}g(z,s)n^{-z/2}+O\left(e^{C\sqrt{n}}{n}^{-\frac{t+3}{2}}\right).$$

$$\sum_{J\subset[t]}(-1)^{|J|}\left(\sum_{i\in J}s_{i}\right)^{z}=\left\{\begin{array}[]{ll}0&\mbox{for }z<t\\ (-1)^{t}t!\prod_{i=1}^{t}s_{i}&\mbox{for }z=t.\end{array}\right.$$

$$[s_{j_1}^{{\alpha }_1}\cdots s_{j_k}^{{\alpha }_k}]f=(-1{)}^k(\genfrac{}{}{0ex}{}{z}{{\alpha }_1,\dots ,{\alpha }_k})\sum _{l=0}^{t-k}(-1{)}^l\left(\genfrac{}{}{0ex}{}{t-k}{l}\right)$$

$$[s_1\cdots s_t]f=(-1{)}^t(\genfrac{}{}{0ex}{}{z}{1,\dots ,1})=(-1{)}^{t}z!=(-1{)}^{t}t!,$$

$$p_{-S}(n)=\sum_{J\subset[t]}(-1)^{|J|}p\bigg{(}n-\sum_{j\in J}s_{j}\bigg{)}\;.$$

$$p_{-S}(n)=$$

$$=$$

$$\sum _{J\subset [t]}(-1{)}^{|J|}g(z,\sum _{j\in J}{s}_j)=0.$$

$$\displaystyle p_{-S}(n)=\frac{e^{C\sqrt{n}}}{4\pi n\sqrt{2}}\sum_{J\subset[t]}(-1)^{|J|}g\bigg{(}t,\sum_{j\in J}s_{j}\bigg{)}n^{-t/2}+O\left(e^{C\sqrt{n}}n^{-\frac{t+3}{2}}\right).$$

$$\displaystyle\sum_{J\subset[t]}(-1)^{|J|}\frac{(-1)^{t}C^{t+1}}{2^{t}t!}\bigg{(}\sum_{j\in J}s_{j}\bigg{)}^{t}=\frac{(-1)^{t}C^{t+1}}{2^{t}t!}(-1)^{t}t!\prod_{i=1}^{t}s_{i}=\frac{C^{t+1}}{2^{t}}\prod_{i=1}^{t}s_{i}\;.$$

Full text

1 Introduction

Let $n\in\mathbb{N}$. We say that $\lambda=(\lambda_{1},\dots,\lambda_{k})$, where $\lambda_{1}\geq\dots\geq\lambda_{k}>0$ are positive integers, is a partition of $n$ if $\sum_{i=1}^{k}\lambda_{i}=n$. We say that $k$ is the length of $\lambda$ and $\lambda_{1},\dots,\lambda_{k}$ are the parts of the partition $\lambda$.

Let $p(n)$ be the number of partitions of $n$. The famous result of Hardy and Ramanujan [5] gives the asymptotics of $p(n)$ as

[TABLE]

As usual, $f(n)\sim g(n)$ means that $\lim_{n\to\infty}\frac{f(n)}{g(n)}=1$. For more information on partitions refer to the book [1] of G. Andrews. In [6] F. Johansson presents an almost optimum numerical algorithm for evaluating $p(n)$.

One can be also interested in the number $p_{S}(n)$ of partitions of $n$ with parts from a given (finite or infinite) set $S\subset\mathbb{N}$. Such partitions are often called “restricted partitions”. I. Schur [9] proved that for finite $S$ with cardinality $|S|=t$ (and such that the $\mathrm{gcd}(S)=1$) one has the asymptotics

[TABLE]

Note that adding a new element $s$ to $S$ increases $p_{S}(n)$ by the factor $\frac{n}{s|S|}$.

Motivated by these two partition asymptotics, we obtain what we call a complementary Schur asymptotic. Nicolas and Sárközy [7] found an asymptotics of the number of partitions using only parts $\geq m$ for a wide range of parameter $m$. We extend this result. For a finite set of integers $S\subset\mathbb{N}$, $|S|=t$, let $p_{-S}(n)$ be the number of partitions of $n$ not using any part from $S$. Our main result is that

[TABLE]

Now each element $s$ in $S$ decreases $p(n)$ by the factor $\frac{Cs}{2\sqrt{n}}$. We apply this to construct highly oscillating partition ideals. This theorem is an extension of result [7] of Nicolas and Sárközy who considered the case $S=[m]$.

The paper is organized as follows. In Section 2 we state the strong Hardy–Ramanujan asymptotics of $p(n)$ and restate our main result. Section 3 presents some useful lemmas. Section 4 is devoted to the proof of the main theorem. In the last Section 5 we present an application of the main result to oscillations of growth functions of partition ideals.

2 Two asymptotics for partitions

Let $n\in\mathbb{N}$,

[TABLE]

be a constant.

As we mentioned earlier, the classical asymptotics for $p(n)$ proved by Hardy and Ramanujan [5] is $p(n)\sim e^{C\sqrt{n}}/4n\sqrt{3}$. In [5, formula (1.55)] they gave much stronger asymptotics which we give in the next theorem. They state it as a value of a derivative which we compute explicitly.

Theorem 1** (Hardy–Ramanujan).**

For $n=1,2,\dots$, the partition function $p(n)$ satisfies

[TABLE]

Note that now the error is only about a square root of the main term. This resembles strong asymptotic relations for coefficients of power series with unique dominant singularity (see P. Flajolet and R. Sedgewick [3, Chapter V] or think of the Fibonacci numbers). We use this result with small error term to deduce our asymptotic relation for $p_{-S}(n)$. Our main theorem says:

Theorem 2**.**

Let $S=\{s_{1},s_{2},\dots,s_{t}\}\subset\mathbb{N}$ be a finite set of integers with $|S|=t$. Then the number $p_{-S}(n)$ of partitions of $n$ with parts in $\mathbb{N}\backslash S$ satisfies

[TABLE]

We base our proof on manipulating the strong asymptotics of $p(n)$. First we estimate $p(n-s)$ for a fixed $s\in\mathbb{N}$, and then express $p_{-S}(n)$ as a sum of values $p(n-s)$ for various numbers $s$.

3 Auxiliary results

Let $S\subset\mathbb{N}$ be a finite set with $|S|=t$ and let $s\in[t]$. First we determine the asymptotics of $p(n-s)$ and then we prove an algebraic identity needed later in the proof of the main result.

We set

[TABLE]

Lemma 1**.**

Let $t\in\mathbb{N}$. Then for all $n,s\in\mathbb{N}$ with $n>s$ we have

[TABLE]

where for $z\in\{0,1,\dots,t\}$ we denote by $g(z,s)$ a real polynomial in $s$ with degree $z$ and leading term

[TABLE]

where $h(z,s)$ is a real polynomial in $s$ with degree at most $z-1$.

Proof.

Let $t\in\mathbb{N}$. We expand the main term in Theorem 1 in powers of $n$. Since $e^{x}=\sum_{i\geq 0}\frac{x^{i}}{i!}=\sum_{i=0}^{l}\frac{x^{i}}{i!}+O(x^{l+1})$ for $|x|<c<1$ and ($c_{i}$ are some real constants, not necessarily always the same)

[TABLE]

we get, for every $n>n_{0}(t)$ and some coefficients $a_{k}$,

[TABLE]

(note that $a_{0}=C$). For all integers $n>s>0$ we have, expanding again $(n-s)^{-k/2-1}=n^{-k/2-1}(1-s/n)^{-k/2-1}=\dots$ by I. Newton’s binomial theorem, that

[TABLE]

where

[TABLE]

is a real polynomial in $s$ with $\deg_{s}f(w,s)=\lfloor w/2\rfloor$.

Next we combine expansions of the square root and of the exponential function, as above, and denote $x=n^{-1/2}$. For all integers $n>s>0$, where $s$ is fixed, we get

[TABLE]

where $d(i,s)$ is a polynomial in $s$ with $\deg_{s}d(i,s)=i$ and

[TABLE]

($c_{i-1}s^{i-1}+\dots$ are the remaining terms with degree less than $i$).

Combining results (4), (5) and (7) we have

[TABLE]

where

[TABLE]

is a real polynomial in $s$ with $\deg_{s}g(z,s)=z$. Indeed, $\max_{i,w\geq 0,\,i+w=z}(i+\lfloor w/2\rfloor)=z$, attained uniquely for $i=z$ and $w=0$. Moreover, (6) and (8) imply

[TABLE]

where $h(z,s)$ are the remaining terms with $\deg_{s}h(z,s)\leq z-1$. The exponent of $C$ increased by one since $a_{0}=C$. Therefore for all integers $n>s>0$,

[TABLE]

∎

We state and prove an identity needed in the main proof. Recall that for $t\in\mathbb{N}_{0}$, $[t]=\{1,2,\dots,t\}$.

Lemma 2**.**

Let $t\geq z\geq 0$ be integers and $s_{1},s_{2},\dots,s_{t}$ be variables. Then

[TABLE]

The first case holds in fact more generally for any polynomial in $\sum_{i\in J}s_{i}$ with degree at most $t-1$.

Proof.

Let $k\leq t$ be positive integers, $j_{i}$ with $1\leq j_{1}<j_{2}<\dots<j_{k}\leq t$ be some indices and $\alpha_{1},\dots,\alpha_{k}$ be positive integers with $\sum\alpha_{i}=z$, so $k\leq z$. We denote the polynomial on the left side as $f=f(s_{1},\dots,s_{t})$ and examine the coefficient $[s_{j_{1}}^{\alpha_{1}}\cdots s_{j_{k}}^{\alpha_{k}}]f$. Clearly, only the sets $J$ containing $\{j_{1},\dots,j_{k}\}$ contribute to it. Each such $J$ contributes $(-1)^{|J|}\binom{z}{\alpha_{1},\dots,\alpha_{k}}$ to the coefficient. Summing over all $J$ containing $\{j_{1},\dots,j_{k}\}$ we obtain

[TABLE]

(here $l=|J\backslash\{j_{1},\dots,j_{k}\}|$). If $z<t$ then $k<t$ and the sum is $(1-1)^{t-k}=0$ by the binomial theorem. If $z=t$ then only $k=z=t$ yields non-zero contribution, for $\alpha_{1}=\dots=\alpha_{k}=1$, and we get the coefficient

[TABLE]

which proves the theorem. ∎

4 Proof of the Theorem 2

Proof of Theorem 2.

Let $J\subset[t]$. Note that the partitions of $n$ using each part $s_{j}$, $j\in J$, at least once are in bijection with the partitions of $n-\sum_{j\in J}s_{j}$. Thus by the principle of inclusion and exclusion,

[TABLE]

By Lemma 1 we have

[TABLE]

We apply Lemma 2 to the polynomial $g(z,\sum_{j\in J}s_{j})$ when $z\in[t-1]$—first we understand $s_{j}$ as variables and only at the end we substitute for them the numbers $s_{j}$—and obtain, by the first case,

[TABLE]

Hence only term $z=t$ remains yielding

[TABLE]

Finally, we expand $g(t,\sum s_{j})$ by equation (3) and use Lemma 2. By the first case of Lemma 2 the contributions of $h(z,\sum s_{j})$ sum up to zero. By the second case the contribution of the leading term in $g(z,\sum s_{j})$ is

[TABLE]

Thus

[TABLE]

which in view that $C=\pi\sqrt{2/3}$ and $p(n)\sim e^{C\sqrt{n}}/4n\sqrt{3}$ gives the desired asymptotics

[TABLE]

∎

5 Partition functions of ideals

Let $X$ be a set of partitions and $\lambda,\gamma\in X$. We say that $\lambda$ is a subpartition of $\gamma$ if no part from $\lambda$ has more occurrences in $\lambda$ then in $\gamma$. We denote this relation $\lambda<\gamma$. A set of partitions $X$ is a partition ideal, if $\lambda<\gamma$ and $\gamma\in X$ always implies $\lambda\in X$. By $p(n,X)$ we denote the number of partitions of $n$ lying in $X$. Let $Z$ be a set of partitions such that no two partitions in $Z$ are comparable by $<$. We denote by $F_{Z}$ the set of all partitions that do not contain any element of $Z$. Clearly, $F_{Z}$ is a partition ideal. We call $Z$ a basis of the ideal $F_{Z}$. We denote its counting function by

[TABLE]

Recall that the notation $p_{-S}(n)$ and $p_{-Z}(n)$ where $S$ is a set of positive integers and $Z$ is a set of partitions. Finally, two partitions are independent if their parts are pairwise distinct.

We make use of the Cohen–Remmel theorem [2, 8] that gives sufficient condition for equality of counting functions of two partition ideals in terms of their bases. In the theorem we use the following notation. For a partition $\lambda$, $|\lambda|$ is the sum of all parts of $\lambda$, and for several partitions $\lambda^{i}$ their union is the partition $\lambda$ such that the multiplicity of any part equals to the maximum multiplicity attained over all $\lambda^{i}$.

Theorem 3** (Cohen, 1981; Remmel, 1982).**

Let $\Lambda=\{\lambda^{1},\lambda^{2},\dots\}$ and $\Gamma=\{\gamma^{1},\gamma^{2},\dots\}$ be two finite or infinite sequences of partitions (of the same length), such that for every finite set $I\subset\mathbb{N}$,

[TABLE]

Then $p_{-\Lambda}(n)=p_{-\Gamma}(n)$ for every positive integer $n$.

Now we state three applications of our Theorem 2. We are inspired by similar results of Hančl [4], but use a different approach.

Theorem 4**.**

Let $F_{Z}$ be a partition ideal with basis $Z$, where $Z$ contains infinitely many pairwise independent partitions, and let $k$ be any positive integer. Then

[TABLE]

for any sufficiently large $n$, where $K=K(k,Z)$ is a fixed constant (and $C=\pi\sqrt{2/3}$).

Proof.

Let $k\in\mathbb{N}$ and $t=2k-1$. Let $\lambda^{1},\lambda^{2},\dots,\lambda^{t}$ be mutually independent partitions from $Z$ such that $|\lambda^{1}|<|\lambda^{2}|<\cdots<|\lambda^{t}|$. By Theorem 3, applied to $\Lambda=\{\lambda^{1},\lambda^{2},\dots,\lambda^{t}\}$ and $\Gamma=\{(|\lambda^{1}|),(|\lambda^{2}|),\dots,(|\lambda^{t}|)\}$,

[TABLE]

From Theorem 2 we have

[TABLE]

where $K=K(k,Z)$ is a constant and the last inequality holds for any sufficiently large $n$. ∎

Conjecture 1**.**

Let $X=F_{Z}$ be a partition ideal with finite basis $Z$. Then the asymptotics for the counting function $p(n,X)$ is of the form

[TABLE]

where $K$ is a constant and $k=m/2$ for some $m\in\mathbb{N}$.

Let $\varepsilon\in(0,1)$. For the forthcoming theorem we set

[TABLE]

Thus $f(n)$ goes to $1$ as fast as $n^{-1/2}\log^{1+\varepsilon}n$ goes to [math].

Theorem 5**.**

Let $\varepsilon>0$ and $f(n)$ be as above. Then there is a partition ideal $X$ such that both

$p(n,X)=0$** 2. 2.

$p(n,X)>p(nf(n))$**

holds for infinitely many positive integers $n$.

Proof.

We define the sequences $(s_{i})_{i=1}^{\infty}$ and $(t_{i})_{i=1}^{\infty}$ of positive integers such that $s_{1}=2$,

[TABLE]

and, given $s_{i}$, we set

[TABLE]

where $n_{0}=n_{0}(s_{i})$ is such that for any $n\geq n_{0}$ we have $f(n)\in(\frac{1}{2},1)$ and both

[TABLE]

Existence of $n_{0}$ follows from the Hardy–Ramanujan asymptotics (1) and Theorem 2.

Let $I_{i}=[s_{i},t_{i}]\cap\mathbb{N}$. Let $X$ be the partition ideal consisting of the partitions that use parts from any of the intervals $I_{i}$ with multiplicities at most $t_{i}$, and do not use other parts. Our aim is to prove that the first condition is satisfied for $n=s_{i+1}-1$ and the second condition is satisfied for $n=t_{i+1}$.

Any partition of $s_{i+1}-1$ lying in $X$ may use only parts $\leq t_{i}$ but, as the multiplicities are restricted, sum of all the parts $\leq t_{i}$ equals

[TABLE]

Hence easily $p(s_{i+1}-1,X)=0$ for any positive integer $i$.

Let $K=(8\sqrt{3})^{-1}$. To prove the second property we first show that for a fixed positive integer $i$ and any $n>\max\{t_{i},n_{0}\}$ we have

[TABLE]

Indeed, the definition of $f(n)$ implies $\sqrt{n}(1-\sqrt{f(n)})=\log^{1+\varepsilon}n$ and $0<f(n)<1$, which, combined with the definition of $t_{i}$ and $n_{0}$, yields for any $n\geq\max\{t_{i},n_{0}\}$ bound

[TABLE]

by (9) and the bounds $K>\frac{1}{16}$ and $n\geq 2$. Now Theorem 2 and (10) implies

[TABLE]

We apply again asymptotics (1) for $p(t_{i})$ and $p(t_{i}f(t_{i}))$ and have

[TABLE]

Putting these results together we get that

[TABLE]

since $s_{i}\geq 2,Ct_{i}\geq 6$. That completes the proof. ∎

6 Acknowledgments

I would like to thank my supervisor Martin Klazar for useful conversations and corrections. The work was supported by the grant SVV–2017–260452 and SVV-2017-260456.