### FERRERS MATRICES AND RELATED PARTITIONS OF INTEGERS

This page is a supplement for certain sequences submitted to the Online Encyclopedia of Integer Sequences on February 26, 2014, by

Clark Kimberling
University of Evansville

and

Peter Moses
Moparmatic Co.
Redditch, Worcestershire, UK

Key words for this work are partitions and Ferrers graphs (also called Ferrers diagrams and Ferrers plots) of partitions. Definitions of these terms and introductions to various kinds of partitions can be found at MathWorld. Here, we define the Ferrers matrix of a partition and use it as a basis for defining seven kinds of partitions which seem to be new: NW, NE, SE, SW, square, antidiagonal, and diagonal.

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