Lecture hall position conjecture for the Euler partition bijection

Proof for every odd partition. Let $\lambda$ be a nonempty partition into odd parts, let its largest part occur $r$ times, and write

$$\pi=\varphi(\lambda) =(\pi_1>\cdots>\pi_k>0), \qquad \pi_{k+1}=0, \qquad m_j=\pi_j-\pi_{j+1}.$$

By source Theorem 4.2, its lecture-hall length is

$$L=\ell_h(\pi) = \max_{1\le j\le k} \left( \left\lceil\frac{\pi_j}{m_j}\right\rceil+j-1 \right).$$

Define the maximizing index set

$$I= \left\{ j: \left\lceil\frac{\pi_j}{m_j}\right\rceil+j-1=L \right\}.$$

For $j\in I$, write uniquely

$$\pi_j=(L-j)m_j+r_j, \qquad 1\le r_j\le m_j.$$

The source defines the lecture-hall position by choosing a maximizing index with the largest $r_j$, resolving any remaining tie by the smallest index. Thus

$$r_\pi=\max_{j\in I}r_j. \tag{1}$$

Let $\tau$ be the source's descent on odd partitions. Its gap-free coding sequence is

$$\bigl( b_o(\lambda), b_o(\tau\lambda), b_o(\tau^2\lambda), \ldots \bigr),$$

and $\varphi(\lambda)$ is the conjugate partition of that sequence. Applying $\tau$ deletes its first entry, hence removes the first column of the conjugate:

$$\varphi(\tau^s\lambda) = (\pi_1-s,\ldots,\pi_k-s)_{>0} =:\pi^{(s)}. \tag{2}$$

Source Theorem 7.6 proves, for every nonempty odd partition $\mu$,

$$g(\mu)=2\ell_h(\varphi(\mu))-1, \tag{3}$$

where $g(\mu)$ is its largest part. Each descent replaces exactly one occurrence of the current largest odd part $t$ by $t-2$, or deletes one part if $t=1$. Therefore the original largest part remains unchanged for exactly $r$ successive states. By (2)–(3),

$$r= \min\{s\ge1:\ell_h(\pi^{(s)})<L\}, \tag{4}$$

with $\ell_h(\varnothing)=0$.

It remains to compute this first-drop time intrinsically. First $L\ge k$, since the contribution of $j=k$ is $k$. Whenever $j$ remains nonterminal after subtracting $s$, its gap is unchanged and its contribution is

$$F_j(s) = \left\lceil\frac{\pi_j-s}{m_j}\right\rceil+j-1.$$

If $j\in I$ and $j<k$, then

$$\pi_{j+1} = (L-j-1)m_j+r_j \ge r_j.$$

Thus for $0\le s<r_j$, the index remains nonterminal and

$$F_j(s) = L-1+ \left\lceil\frac{r_j-s}{m_j}\right\rceil =L.$$

For $s\ge r_j$, its contribution is at most $L-1$ while nonterminal; if it becomes terminal, its contribution is $j<L$.

The index $j=k$ can belong to $I$ only when $L=k$, in which case $r_k=\pi_k$. Its contribution remains $L$ precisely for $s<r_k$ and disappears afterward. An initially nonmaximizing index never becomes maximizing: its nonterminal contribution is nonincreasing, and a terminal contribution is $j<L$, unless $j=k=L$, which was already a maximizing index.

Consequently,

$$\ell_h(\pi^{(s)})=L \quad\Longleftrightarrow\quad s<\max_{j\in I}r_j.$$

Combining this with (1) and (4) gives

$$\boxed{r=r_\pi.}$$

This proves Conjecture 8.2 for every odd partition, including partitions consisting entirely of ones.

Source: Wang and Zhang, Euler's partition theorem and lecture hall partition theorem, Conjecture 8.2, https://arxiv.org/abs/2607.16690.