Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it.


QUESTION:


Let $P\_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ parts bounded by $N.$ Prove that, if $n$ and $N$ are sufficiently large, $$P\_{n,N}(k)\sim N^{n-1}\sqrt{\dfrac{6}{\pi n}}\exp{\left[-\frac{3}{2n}\left(\frac{nN+n-2k}{N}\right)^{2}\right]}.$$


PROLOGUE:


Following a suggestion given by Daniele Tampieri in [his answer](https://mathoverflow.net/questions/102597/autobiographies-of-mathematicians/376423#376423) to [this question](https://mathoverflow.net/questions/102597/autobiographies-of-mathematicians/376423%7Bdb005102-8faa-494e-b99a-769c1b99bc26%7D376423) (I acknowledge Daniele for this inspiring advice), I read the autobiographical work of F. Tricomi [1] where the author describes, together with some autobiographical facts, some of his works. In particular he speaks about his paper [2]. Tricomi says that, in this paper, he considers the number $P\_{n,N}(k)$ of composition of an integer $k$ in $n$ parts bounded above by $N$ (I retained the same notations used by Tricomi) and that he proves that, asymptotically, if $n$ and $N$ are sufficiently large, the above estimate holds. That formula reminds me the Hardy-Ramanujan-Rademacher formula for the number of partitions of an integer.


I found the description of this paper a little bit surprising because Tricomi is best known for his work on ordinary differential equations and special function, not for works on number theory/combinatorics.


Since I am not able to find a copy of the original paper of Tricomi, my question is how to prove his asymptotic formula.


WHAT I TRIED TO DO:


I write down the generating function $$\sum\_k P\_{n,N}(k)x^k$$ which is trivially $$\left(\frac{x-x^{N+1}}{1-x}\right)^{n}$$ from which one can find the explicit expression $$P\_{n,N}(k)=\sum\_{j}(-1)^j\binom{n}{j}\binom{k-1-Nj}{n-1}.$$
Then I tried to apply the classical result for the asymptotic estimates in analytic combinatorics (see e.g. [3]) without success.


Perhaps some asymptotic for large binomial coefficient of which I am not aware is necessary?


REFERENCES


[1] Tricomi, Francesco; La mia vita di matematico attraverso la cronistoria dei miei lavori. (Bibliografia commentata 1916–1967), Padova: CEDAM – Casa Editrice Dottor Antonio Milani, pp. XII+172 (1967), MR0274255, Zbl 0199.28603.


[2] Tricomi, Francesco; Su di una variabile casuale connessa con un notevole tipo di partizioni di un intero. Giornale dell'Istituto Italiano degli Attuari, 2, 455-468 (1931), [JFM 57.0614.02](https://zbmath.org/57.0614.02), [Zbl 0003.01703](https://zbmath.org/0003.01703).


[3] Flajolet, Philippe; Sedgewick, Robert; *Analytic combinatorics.* Cambridge University Press, Cambridge, 2009. xiv+810 pp., MR2483235, Zbl 1165.05001.



Proof of an asymptotic formula by Tricomi
$\newcommand{\si}{\sigma}
\newcommand{\Z}{\mathbb Z}$As noted in [Pietro Majer's](https://mathoverflow.net/questions/443268/proof-of-an-asymptotic-formula-by-tricomi#comment1144269_443268) comment, the meaning of $\sim$ in the claim that, if $n$ and $N$ are sufficiently large, then
\begin{equation\*}
 P\_{n,N}(k)\sim N^{n-1}\sqrt{\frac6{\pi n}}
 \exp\Big(-\frac3{2n}\Big(\frac{nN+n-2k}N\Big)^2\Big) \tag{1}\label{1}
\end{equation\*}
should be clarified.
Looking at the constant factor $\sqrt{\frac6\pi}$ on the right-hand side of \eqref{1}, one should apparently assume that $A\sim B$ is supposed to mean here that $A/B\to1$ (as it is common). So, then the claim is that \eqref{1} holds uniformly in $k$ (or at least for each feasible value of $k$) as $n,N\to\infty$.


Then \eqref{1} will of course be false. For instance, if $k=n$, then $P\_{n,N}(k)=1$, whereas the right-hand side of \eqref{1} goes to $\infty$ (super-exponentially fast).


Below we will see for what values of $k$ the asymptotic relation \eqref{1} holds. We will also see that a slight modification of \eqref{1} holds for finite $N\ge2$ and such values of $k$.




---


The proper setting for statements like \eqref{1} is the so-called local (central) limit theorems (LLT's) of probability theory. Indeed, let
\begin{equation\*}
 S\_{n,N}:=X\_{1,N}+\dots+X\_{n,N},
\end{equation\*}
where for each $N$ we have that $X\_{1,N},\dots,X\_{n,N}$ are independent random variables (r.v.'s) each uniformly distributed on the set $[N]:=\{1,\dots,N\}$. Then
\begin{equation\*}
 P\_{n,N}(k)=N^n P(S\_{n,N}=k). \tag{2}\label{2}
\end{equation\*}
Moreover, one has the following LLT:  

\begin{equation\*}
 P(S\_{n,N}=k)=\frac1{\si\_N\sqrt{2\pi n}} e^{-z\_k^2/2}
 +o\Big(\frac1{\si\_N\sqrt n}\Big) \tag{3}\label{3}
\end{equation\*}
uniformly in $k\in\Z$, where $z\_k:=z\_{k;n,N}:=\dfrac{k-n\mu\_N}{\si\_N\,\sqrt n}$,
$\mu\_N:=EX\_{1,N}=(N+1)/2$, and $\si\_N:=\sqrt{Var\,X\_{1,N}}=\sqrt{(N^2-1)/12}$. (The condition $N\to\infty$ is not needed. One may also note that the case $N=2$ of \eqref{3} is also the case $p=q$ of the de Moivre–Laplace theorem.)



> 
> It follows immediately from \eqref{2} and \eqref{3} that \eqref{1} holds as $n,N\to\infty$ uniformly in all values $k$ that are in the normal deviation zone from the mean $n\mu\_N$ of $S\_{n,N}$ -- that is, uniformly in $k$ such that $|z\_k|=O(1)$. Moreover, the slight modification of \eqref{1} for finite $N\ge2$ -- with $N^{n-1}$ replaced by $N^n/\sqrt{N^2-1}$ and $\dfrac{nN+n-2k}N$ replaced by $\dfrac{nN+n-2k}{\sqrt{N^2-1}}$ --
> holds, again uniformly in $k$ such that $|z\_k|=O(1)$.
> 
> 
> 


The proof of \eqref{3} is similar to the proof of, say, Theorem 1 in Ch. VII of [Petrov's book](https://link.springer.com/book/10.1007/978-3-642-65809-9). That theorem is stated for a sequence $(X\_j)$ of (lattice-valued) r.v.'s, whereas in our case we have to deal with a double-index array $(X\_{j,N})$. However, the $X\_{j,N}$'s have the specific and easy to deal with uniform distribution on the set $[N]$, which makes the proof of \eqref{3} significantly easier overall than the proof of the mentioned theorem. Indeed, in view of (say) Lemma 2 of [Gamkrelidze](https://epubs.siam.org/doi/10.1137/1133053), we can deal with the most difficult integral $I\_3$ in the proof of the mentioned theorem the same way as with the easy integrals $I\_2$ and $I\_4$ (the four integrals $I\_1,\dots,I\_4$ are defined at the bottom of p. 190 of Petrov's book).


