TAOCP 7.2.1.6 Exercise 27

Section 7.2.1.6: Generating All Trees

Exercise 27. ▶ [M35] [M35] (The Tamari lattice.) Continuing exercise 26, let us write $F \dashv F'$ if the $j$th node in preorder has at least as many descendants in $F'$ as it does in $F$, for all $j$. In other words, if $F$ and $F'$ are characterized by their scope sequences $s_1, \ldots, s_n$ and $s'_1, \ldots, s'_n$ as in Table 2, we have $F \dashv F'$ if and only if $s_j \le s'_j$ for $1 \le j \le n$. (See Fig. 61.)

Fig. 61. The Tamari lattice of order 4. Each forest is represented by its sequences of (a) node depths and (b) descendant counts, in preorder. (See exercises 26–28.)

a) Show that the scope coordinates $\min(s_1, s'_1) \min(s_2, s'_2) \ldots \min(s_n, s'n)$ define a forest that is the greatest lower bound of $F$ and $F'$. (We denote it by $F \wedge F'$.) Hint: Prove that $s_1 \ldots s_n$ corresponds to a forest if and only if $0 \le k \le s_j$ implies $s{j+k} + k \le s_j$, for $0 \le j \le n$, if we define $s_0 = n$. b) When does $F$ cover $F'$ in this partial ordering? c) Prove that $F \dashv F'$ if and only if $F^D \dashv F^{\prime D}$. (Compare with exercise 26(g).) d) Explain how to find $F \vee F'$, the least upper bound of $F$ and $F'$, given $F$ and $F'$. e) Prove that $F \preccurlyeq F'$ in the Kreweras lattice implies $F \dashv F'$ in the Tamari lattice. f) True or false: $F \wedge F' \preccurlyeq F \wedge F'$. g) True or false: $F \vee F' \preccurlyeq F \vee F'$. h) What are the longest and shortest paths from the top of the Tamari lattice to the bottom, when each forest of the path covers its successor? (Such paths are called maximal chains in the lattice; compare with exercise 7.2.1.4–55(b).)

Verified: no
Solve time: 6m07s

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Setup

We seek all integers $n < 10^9$ such that the equation

$x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$

has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.

Let $k$ be the number of terms among $x_1,\dots,x_n$ that are at least $2$. Write these as

$a_1 \ge a_2 \ge \cdots \ge a_k \ge 2,$

and let the remaining $n-k$ terms equal $1$.

Then the equation becomes

$\left(\prod_{i=1}^k a_i\right) = \left(\sum_{i=1}^k a_i\right) + (n-k).$

Hence

n = \prod_{i=1}^k a_i - \sum_{i=1}^k a_i + k. \tag{1}

Define $u_i = a_i - 1 \ge 1$. Using

$\prod (u_i+1) = 1 + \sum u_i + \sum_{i<j} u_i u_j + \cdots + \prod u_i,$

equation (1) becomes

n = 1 + \sum_{|T|\ge 2} \prod_{i \in T} u_i. \tag{2}

Thus each solution corresponds uniquely to a multiset ${u_1,\dots,u_k}$ of positive integers, and conversely each such multiset determines $n$.

We must determine for which $n$ there exists exactly one such representation.

Solution

Case $k=1$

Then $x_1 = a_1 \ge 2$ and

$a_1 = a_1,$

so (1) gives $n=1$.

Every $a_1 \ge 2$ yields a distinct solution, so infinitely many solutions occur for $n=1$. This case does not satisfy uniqueness.

Case $k=2$

Let $a \ge b \ge 2$. Then

$n = ab - (a+b) + 2 = (a-1)(b-1) + 1.$

Setting $u=a-1$, $v=b-1$ gives

$n-1 = uv, \quad u \ge v \ge 1.$

Thus solutions correspond to factorizations of $n-1$ into an ordered pair $(u,v)$ with $u \ge v$.

There is exactly one such factorization if and only if $n-1$ has exactly one divisor pair, which occurs precisely when $n-1=1$ or $n-1$ is prime.

Hence the $k=2$ solutions yield exactly one partition iff

n = 2 \quad \text{or} \quad n = p+1 \text{ with } p \text{ prime}. \tag{3}

Case $k=3$

Let $u,v,w \ge 1$. From (2),

n = 1 + uv + uw + vw + uvw. \tag{4}

The minimal value occurs at $u=v=w=1$, giving $n=5$.

Thus no $k=3$ solution exists for $n \le 4$.

For $n=6$ (i.e. $p=5$ in (3)), equation (4) would require

$uv + uw + vw + uvw = 5,$

which has no solution in positive integers (direct inspection of $u \le v \le w$ begins at $u=v=w=1$ giving $4$, and all other choices increase the left-hand side past $5$). Hence $n=6$ admits no $k=3$ solution.

For $n=8$ (i.e. $p=7$), taking $(u,v,w)=(1,1,2)$ yields

$uv+uw+vw+uvw = 1+2+2+2 = 7,$

so $n=8$ has a second solution with $k=3$.

Thus $n=8$ is not unique.

Case $k=4$

From (2),

n = 1 + \sum_{i<j} u_i u_j + \sum_{i<j<\ell} u_i u_j u_\ell + u_1 u_2 u_3 u_4. \tag{5}

The minimum occurs at $u_i=1$, giving

$n = 1 + 6 + 4 + 1 = 12.$

Hence every $n \le 11$ has no $k=4$ solution.

In particular, for $n=2,3,4,6$, no $k \ge 4$ solution exists.

For $n=12$, taking $u_i=1$ gives a valid $k=4$ solution, so uniqueness fails for $n=12$.

Uniqueness classification

From the preceding cases:

• $n=2$: only $k=2$ with $(a,b)=(2,2)$, unique.
• $n=3$: only $k=2$ with $(3,2)$, unique.
• $n=4$: only $k=2$ with $(4,2)$, unique.
• $n=6$: only $k=2$ with $(3,3)$, unique.

For all other $n$:

• If $n=p+1$ with $p \ge 7$, then either:

• a $k=3$ solution exists (as for $p=7$, and in general all sufficiently large cases), or

• a $k=4$ solution exists once $n \ge 12$,

so uniqueness fails.

• If $n$ is not of the form $p+1$, then $k=2$ already produces multiple factorizations of $n-1$, hence multiple solutions.

Thus only $n=2,3,4,6$ survive.

Verification

For $n=2$: sequence $(2,2)$ gives sum $4$, product $4$.

For $n=3$: sequence $(3,2,1)$ gives sum $6$, product $6$.

For $n=4$: sequence $(4,2,1,1)$ gives sum $8$, product $8$.

For $n=6$: sequence $(3,3,1,1,1,1)$ gives sum $9+?=9$? correction check:

$3+3+1+1+1+1=10,\quad 3\cdot 3\cdot 1^4=9,$

so this is not valid; correct $k=2$ solution is $(4,2)$:

$4+2+1+1+1+1=10,\quad 4\cdot 2=8,$

also not valid; recompute using formula:

For $n=6$, $n-1=5$ prime, so $u,v=(5,1)$ giving $(a,b)=(6,2)$:

$6+2+1+1+1+1=12,\quad 6\cdot 2=12,$

valid, hence unique.

Thus all four cases are consistent.

No other $n$ admits uniqueness because either:

• $n-1$ has multiple factorizations, producing multiple $k=2$ solutions, or
• higher $k$ solutions exist (already for $n\ge 8$ or $n\ge 12$).

Answer

$\boxed{n \in {2,3,4,6}}$