Clark Kimberling

University of Evansville

and

Peter Moses

Moparmatic Co.

Redditch, Worcestershire, UK

A *Ferrers matrix* can be defined **without** reference to *Ferrers graph* as an mXm matrix (x(i,j)) of 0s and 1s satisfying these conditions:

(1) x(1, m) = 1 or x(m, 1) = 1;

(2) x(i, j+1) >= x(i, j) for i = 1, 2, ..., m and j = 1, 2, ..., m-1;

(3) x(i+1, j) >= x(i, j) for i = 1, 2, ..., m-1 and j = 1, 2, ..., m.

To see the connection between a Ferrers matrix and a Ferrers graph, suppose that p is a partition (of some n), and let m = max{greatest part of p, number of parts of p}. Write the Ferrers graph of p with 1s as nodes, and pad the graph with 0s. The result is the Ferrers matrix of p.

A simple example of a 3X3 Ferrers matrix will serve as a guide to the seven kinds of partitions. Write the matrix as

a b c

d e f

g h i

Writing summands in clockwise order, the four directional partitions of p are given by

NW(p) = [g + d + a + b + c, h + e + f, i]

NE(p) = [a + b + c + f + i, d + e + h, g]

SE(p) = [c + f + i + h + g, b + e + d, a]

SW(p) = [i + h + g + d + a, f + e + b, c]

antidiag(p) = [a, b + d, c + e + g, f + h, i]

diag(p) = [g, d + h, a + e + i, b + f, c]

square(p) = [a + b + c + f + i + h + g + d, e]

The order in which the parts appear does not change the partition, but it is common to list them in nondecreasing order, as in Figure 1, showing the NW, NE, SE, SW, antidiagonal, and diagonal partitions of the partition 5 + 4 + 4 + 2 + 2 + 1 of 18:
**Figure 1**

Sequences (including arrays) defined from the seven kinds of partitions include these:

NW partitions: **A237981**

NE partitions: **A237982**

SE partitions: **A237983**

SW partitions: **A237982**

antidiagonal partitions: **A238325**, **A238004**

diagonal partitions: **A238326**

square partitions: **A237985**, **A237980**