2016 USAMO Problems/Problem 2

Problem

Prove that for any positive integer $k,$ $\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}$ is an integer.

Solution 1

Define $v_p(N)$ for all rational numbers $N$ and primes $p$ , where if $N=\frac{x}{y}$ , then $v_p(N)=v_p(x)-v_p(y)$ , and $v_p(x)$ is the greatest power of $p$ that divides $x$ for integer $x$ . Note that the expression(that we're trying to prove is an integer) is clearly rational, call it $N$ .

$v_p(N)=\sum_{i=1}^\infty \left\lfloor \frac{k^{2}}{p^{i}} \right\rfloor+\sum_{j=0}^{k-1} \sum_{i=1}^\infty \left\lfloor \frac{j}{p^{i}}\right\rfloor-\sum_{j=k}^{2k-1} \sum_{i=1}^\infty \left\lfloor \frac{j}{p^{i}} \right\rfloor$ , by Legendre. Clearly, $\left\lfloor{\frac{x}{p}}\right\rfloor={\frac{x-r(x,p)}{p}}$ , and $\sum_{i=0}^{k-1} r(i,m)\leq \sum_{i=k}^{2k-1} r(i,m)$ , where $r(i,m)$ is the remainder function(we take out groups of $m$ which are just permutations of numbers $1$ to $m$ until there are less than $m$ left, then we have $m$ distinct values, which the minimum sum is attained at $0$ to $k-1$ ). Thus, $v_p(N)=\sum_{m=p^{i}, i\in \mathbb{N}_{+}}-\frac{k^{2}}{m}+\left\lfloor{\frac{k^{2}}{m}}\right\rfloor-\frac{\sum_{i=0}^{k-1} r(i,m)-\sum_{i=k}^{2k-1} r(i,m)}{m} \geq \sum_{m=p^{i}, i\in \mathbb{N}} \left\lceil -\frac{k^{2}}{m}+\lfloor{\frac{k^{2}}{m}}\rfloor\right\rceil \geq 0$ , as the term in each summand is a sum of floors also and is clearly an integer.

Solution 2 (Controversial)

Consider an $k\times k$ grid, which is to be filled with the integers $1$ through $k^2$ such that the numbers in each row are in increasing order from left to right, and such that the numbers in each column are in increasing order from bottom to top. In other words, we are creating an $k\times k$ standard Young tableaux.

The Hook Length Formula is the source of the controversy, as it is very powerful and trivializes this problem. The Hook Length Formula states that the number of ways to create this standard Young tableaux (call this $N$ for convenience) is: $N = \frac{\left(k^2\right)!}{\prod_{1\le i, j\le k}(i+j-1)}.$ Now, we do some simple rearrangement: $N = \left(k^2\right)!\cdot\prod_{j=1}^{k}\prod_{i=1}^{k}\frac{1}{i+j-1} = \left(k^2\right)!\cdot\prod_{j=1}^{k}\frac{\left(j-1\right)!}{\left(j+k-1\right)!}$ $= \left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}.$ This is exactly the expression given in the problem! Since the expression given in the problem equals the number of distinct $k\times k$ standard Young tableaux, it must be an integer, so we are done.

Solution 3 (Induction)

Define $A(k) = \left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}.$ Clearly, $A(1) = 1$ and $A(2) = 2.$

Then $\frac{A(k+1)}{A(k)} = \frac{\left(k^2+2k+1\right)!\cdot\prod_{j=0}^{k}\frac{j!}{\left(j+k+1\right)!}}{\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}}.$ Lots of terms cancel, and we are left with $\frac{A(k+1)}{A(k)} = \frac{(k^2+1)(k^2+2)\cdots(k^2+2k+1)}{2(2k+1)}.$ The numerator has $2k+1$ consecutive positive integers, so one of them must be divisible by $(2k+1).$ Also, there are $2k$ terms left, $k$ of which are even. We can choose one of these to cancel out the $2$ in the denominator. Therefore, the ratio between $A(k+1)$ and $A(k)$ is an integer. By our inductive hypothesis, $A(k)$ is an integer. Therefore, $A(k+1)$ is as well, and we are done.

Note: This is incorrect.

Solution 4

Throughout this proof we work in the polynomial ring $\mathbb Z[q]$ . For any positive integer $n$ define the $q$ -integer and the Gaussian factorial (also called $q$ -factorial) by $[n]_q := 1 + q + q^{2} + \dots + q^{\,n-1} \quad\Longrightarrow\quad [n]_q! := \prod_{i=1}^{n}[i]_q.$ Because each $[i]_q$ is a polynomial of degree $i-1$ with integer coefficients, $[n]_q!$ is a finite polynomial in $q$ , and evaluating at $q=1$ recovers the ordinary factorial: $\lim_{q\to1}[n]_q! \;=\; n!.$ We will form a product/quotient of such Gaussian factorials, show that the resulting expression lies in $\mathbb Z[q]$ , and finally plug in $q=1$ to obtain the desired integer.

We want to prove that every factor $(1-q^{d})$ appears with non–negative exponent in $P_k(q)= [k^{2}]_q!\,\prod_{j=0}^{k-1}\frac{[j]_q!}{[j+k]_q!},$ so that $P_k(q)\in\mathbb Z[q]$ .

For any $n\ge1$ the Gaussian factorial factors as $[n]_q! \;=\; (1-q)^{-n}\prod_{d\ge1}(1-q^{d})^{\lfloor n/d\rfloor}.$ Let $\nu_d(F)$ be the exponent of $(1-q^{d})$ in a polynomial $F$ ; then $\nu_d([n]_q!) = \left\lfloor \frac{n}{d} \right\rfloor .$

Apply this to $P_k(q)$ :

$\nu_d(P_k(q)) = \left\lfloor \frac{k^{2}}{d} \right\rfloor + \sum_{j=0}^{k-1}\left\lfloor \frac{j}{d}\right\rfloor - \sum_{j=0}^{k-1}\left\lfloor \frac{j+k}{d}\right\rfloor .$

Rewrite the last two sums together:

$\nu_d(P_k(q)) = \left\lfloor \frac{k^{2}}{d} \right\rfloor - \sum_{j=0}^{k-1}\left( \left\lfloor \frac{j+k}{d}\right\rfloor - \left\lfloor \frac{j}{d}\right\rfloor \right).$

For each $j$ the difference $\left\lfloor \frac{j+k}{d}\right\rfloor - \left\lfloor \frac{j}{d}\right\rfloor$ is $1$ precisely when a multiple of $d$ lies in the interval $(j,\,j+k]$ , and $0$ otherwise. As $j$ runs from $0$ to $k-1$ , every multiple of $d$ in $\{1,\dots,k^{2}\}$ is counted at most once. Hence the entire sum is at most $\lfloor k^{2}/d\rfloor$ , so $\nu_d(P_k(q)) \;\ge\; 0 \quad\text{for every } d\ge1.$

Because every exponent is non–negative, the denominator in the earlier factorial ratio divides the numerator inside $\mathbb Z[q]$ . Therefore $P_k(q)$ itself lies in $\mathbb Z[q]$ , and evaluating at $q=1$ shows the original product $(k^{2})!\,\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}$ is an integer. ~Lopkiloinm

Note: This solution is not the same as the first solution that is only in the integers and not in an integer polynomial field—a distinction which allows us to use $d$ as all integers between 1 and $k^2$ —not forcing us to use the Legendre $p$ -adic valuation but simply the multiplicity of $1-q^d$ . This is because $\mathbb{Z}[q]$ is considered a graded algebra whereas $\mathbb{Z}$ is a non-graded algebra. It is a good strategy is to convert all numbers into a graded algebra and perform integrality test there than just staying in the non-graded integers.

2016 USAMO (Problems • Resources)
Preceded by Problem 1	Followed by Problem 3
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