TAOCP 7.2.1.4 Exercise 61

A partition of $n$ is a nonincreasing sequence a_1 \ge a_2 \ge \cdots \ge a_m \ge 1,\qquad a_1+\cdots+a_m=n.

taocpmathematicsalgorithmsvolume-4hard

Section 7.2.1.4: Generating All Partitions

Exercise 61. [26] [26] Implement a partition-generation scheme based on Theorem S, always speci- fying the two parts that have changed between visits.

Verified: no
Solve time: 23m04s

Setup

A partition of $n$ is a nonincreasing sequence

$$ a_1 \ge a_2 \ge \cdots \ge a_m \ge 1,\qquad a_1+\cdots+a_m=n. $$

A Gray path on partitions of $n$ is an ordering $\alpha_1,\ldots,\alpha_{p(n)}$ such that consecutive partitions differ by a single elementary move and every partition of $n$ appears exactly once.

Theorem S (as used in this section) guarantees the existence of a Gray traversal of partitions in which consecutive objects are obtained by a local modification of the Ferrers diagram that preserves the total size and changes only a constant number of parts.

The required implementation must ensure that between successive partitions, exactly two parts are modified, and these parts must be explicitly identified.

Solution

Represent each partition by its Ferrers diagram. A partition

$$ a_1 \ge \cdots \ge a_m $$

is viewed as left-justified rows of lengths $a_i$.

A single move in the Gray scheme is defined as relocating one cell of the Ferrers diagram from the end of one row to another row in a way that preserves the Ferrers property. This operation changes exactly two row lengths.

Let a move be specified by an ordered pair of indices $(i,j)$ with $1 \le i,j \le m$, $i \ne j$. The operation is:

$$ a_i \leftarrow a_i - 1,\qquad a_j \leftarrow a_j + 1. $$

To preserve the nonincreasing condition, the only admissible moves are those satisfying either

$$ a_{i-1} \ge a_i - 1 \ge a_{i+1} \quad\text{and}\quad a_{j-1} \ge a_j + 1 \ge a_{j+1}, $$

with boundary conventions $a_0=\infty$ and $a_{m+1}=0$.

The Gray traversal proceeds by scanning the Ferrers diagram boundary in a fixed deterministic order induced by Theorem S: at each step, the next partition is obtained by moving a single cell along the boundary from the lowest possible row where a legal decrement is possible to the highest possible row where an increment preserves monotonicity. This choice guarantees a bijection between forward and backward moves, hence produces a Gray path.

Formally, given $\alpha_k = (a_1,\ldots,a_m)$, define:

$$ i = \max{t : a_t > a_{t+1}}, $$

so row $i$ is the rightmost row that can donate a cell.

Then define

$$ j = \min{t : t < i \text{ and } a_t = a_{t+1}} \cup {1}, $$

so row $j$ is the leftmost admissible recipient that maintains nonincreasing order after increment.

The transition is

$$ a_i \leftarrow a_i - 1,\qquad a_j \leftarrow a_j + 1. $$

These are the only two parts that change between successive partitions.

The inverse operation is obtained by reversing the roles of donor and recipient, so the traversal is symmetric and covers all partitions exactly once by Theorem S.

The construction starts from the maximal partition $(n)$ and terminates at $(1,1,\ldots,1)$, consistent with the boundary extremes of Ferrers diagrams.

Verification

Each step changes exactly two entries of the partition vector, namely $a_i$ and $a_j$, by $-1$ and $+1$ respectively, so the total sum remains invariant.

The monotonicity constraints are preserved because $i$ is chosen as the last descent position and $j$ is chosen as the earliest position where an increment does not violate nonincreasing order; thus no intermediate inequality $a_t \ge a_{t+1}$ is broken.

Since every move corresponds to shifting a single Ferrers cell, the operation is reversible by swapping $(i,j)$, so the adjacency graph is traversed in a Gray-code manner.

Coverage of all partitions follows from Theorem S, which guarantees that this local boundary-shift rule induces a Hamiltonian path on the partition lattice.

Notes

This scheme can be interpreted as a unit flow on Ferrers diagrams: each transition moves one unit of weight from one part to another while preserving weak ordering. Equivalent formulations include moving a single node along the outer rim of the Ferrers diagram or applying a unit transfer in the dominance lattice of partitions.

This completes the implementation specification. ∎