Three-dimensional Ferrers diagrams.

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Section 8. As a detailed example, shown below are the 6 (= 3 factorial) transpositions of a particular three-dimensional Ferrers diagram; each has 71 stars and might be regarded as a compound partition of the number 71. Each transposition is shown layer by layer, lettered with minuscules to indicate the sequence in which they should be stacked. Included for each is a rectangular grid of integers (see plane partition) that would yield the Ferrers diagram. Within a grid, the numbers in every row, and every column, are non-increasing. Zeroes are used for padding.

A geometrical interpretation of transposition is that the axes in a three-dimensional Euclidean space are being permuted. The name of each transposition, consisting of three digits in quote marks, is suggestive of how the axes are rearranged; in some cases there is pure rotation and in other cases, pure reflection.

The Durfee cube and tetrahedron follow directly. Also, the extension to four dimensions is mechanical. Meanwhile, boolean operations can be defined in the obvious manner.

Linear algebra provides an avenue for further research, because each grid of numbers can be regarded as a matrix. Introduced here is the abbreviation NNNI to describe these grids, because they contain Non-Negative numbers, and because each row and each column maintains its numbers in Non-Increasing order. Usefully, the sum and product of two NNNI matrices will also be an NNNI matrix, which will generate a three-dimensional Ferrers diagram of its own.

8:1 — transpose "012" 4 3 3 3 2 1 3 3 3 2 2 0 3 3 3 2 1 0 3 3 2 2 1 0 3 2 2 0 0 0 2 2 2 0 0 0 2 2 1 0 0 0 1 0 0 0 0 0		a   6   5   4   1  4 ✻ ✻ ✻ ✻ 3 ✻ ✻ ✻   3 ✻ ✻ ✻   3 ✻ ✻ ✻   2 ✻ ✻     1 ✻	b   5   5   3  3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 2 ✻ ✻   2 ✻ ✻	c   5   4   3  3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 2 ✻ ✻   1 ✻
d   5   4   2  3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 2 ✻ ✻   2 ✻ ✻   1 ✻	e   3   3   1  3 ✻ ✻ ✻ 2 ✻ ✻   2 ✻ ✻	f   3   3  2 ✻ ✻ 2 ✻ ✻ 2 ✻ ✻	g   3   2  2 ✻ ✻ 2 ✻ ✻ 1 ✻	h   1  1 ✻

8:2 — transpose "021" 6 5 4 1 5 5 3 0 5 4 3 0 5 4 2 0 3 3 1 0 3 3 0 0 3 2 0 0 1 0 0 0	a   4   3   3   3   2   1  6 ✻ ✻ ✻ ✻ ✻ ✻ 5 ✻ ✻ ✻ ✻ ✻   4 ✻ ✻ ✻ ✻     1 ✻	b   3   3   3   2   2  5 ✻ ✻ ✻ ✻ ✻ 5 ✻ ✻ ✻ ✻ ✻ 3 ✻ ✻ ✻
c   3   3   3   2   1  5 ✻ ✻ ✻ ✻ ✻ 4 ✻ ✻ ✻ ✻   3 ✻ ✻ ✻	d   3   3   2   2   1  5 ✻ ✻ ✻ ✻ ✻ 4 ✻ ✻ ✻ ✻   2 ✻ ✻	e   3   2   2  3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 1 ✻
f   2   2   2  3 ✻ ✻ ✻ 3 ✻ ✻ ✻	g   2   2   1  3 ✻ ✻ ✻ 2 ✻ ✻	h   1  1 ✻

8:3 — transpose "102" 4 3 3 3 3 2 2 1 3 3 3 3 2 2 2 0 3 3 3 2 2 2 1 0 3 2 2 2 0 0 0 0 2 2 1 1 0 0 0 0 1 0 0 0 0 0 0 0		a   8   7   5   1  4 ✻ ✻ ✻ ✻ 3 ✻ ✻ ✻   3 ✻ ✻ ✻   3 ✻ ✻ ✻   3 ✻ ✻ ✻   2 ✻ ✻     2 ✻ ✻     1 ✻	b   7   7   4  3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 2 ✻ ✻   2 ✻ ✻   2 ✻ ✻
c   7   6   3  3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 3 ✻ ✻ ✻ 2 ✻ ✻   2 ✻ ✻   2 ✻ ✻   1 ✻	d   4   4   1  3 ✻ ✻ ✻ 2 ✻ ✻   2 ✻ ✻   2 ✻ ✻	e   4   2  2 ✻ ✻ 2 ✻ ✻ 1 ✻   1 ✻	f   1  1 ✻

8:4 — transpose "120" 8 7 5 1 7 7 4 0 7 6 3 0 4 4 1 0 4 2 0 0 1 0 0 0	a   4   3   3   3   3   2   2   1  8 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ 7 ✻ ✻ ✻ ✻ ✻ ✻ ✻   5 ✻ ✻ ✻ ✻ ✻       1 ✻
b   3   3   3   3   2   2   2  7 ✻ ✻ ✻ ✻ ✻ ✻ ✻ 7 ✻ ✻ ✻ ✻ ✻ ✻ ✻ 4 ✻ ✻ ✻ ✻	c   3   3   3   2   2   2   1  7 ✻ ✻ ✻ ✻ ✻ ✻ ✻ 6 ✻ ✻ ✻ ✻ ✻ ✻   3 ✻ ✻ ✻
d   3   2   2   2  4 ✻ ✻ ✻ ✻ 4 ✻ ✻ ✻ ✻ 1 ✻	e   2   2   1   1  4 ✻ ✻ ✻ ✻ 2 ✻ ✻	f   1  1 ✻

8:5 — transpose "201" 6 5 5 5 3 3 3 1 5 5 4 4 3 3 2 0 4 3 3 2 1 0 0 0 1 0 0 0 0 0 0 0	a   8   7   7   4   4   1  6 ✻ ✻ ✻ ✻ ✻ ✻ 5 ✻ ✻ ✻ ✻ ✻   5 ✻ ✻ ✻ ✻ ✻   5 ✻ ✻ ✻ ✻ ✻   3 ✻ ✻ ✻       3 ✻ ✻ ✻       3 ✻ ✻ ✻       1 ✻
b   7   7   6   4   2  5 ✻ ✻ ✻ ✻ ✻ 5 ✻ ✻ ✻ ✻ ✻ 4 ✻ ✻ ✻ ✻   4 ✻ ✻ ✻ ✻   3 ✻ ✻ ✻     3 ✻ ✻ ✻     2 ✻ ✻	c   5   4   3   1  4 ✻ ✻ ✻ ✻ 3 ✻ ✻ ✻   3 ✻ ✻ ✻   2 ✻ ✻     1 ✻	d   1  1 ✻

8:6 — transpose "210" 8 7 7 4 4 1 7 7 6 4 2 0 5 4 3 1 0 0 1 0 0 0 0 0	a   6   5   5   5   3   3   3   1  8 ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ 7 ✻ ✻ ✻ ✻ ✻ ✻ ✻   7 ✻ ✻ ✻ ✻ ✻ ✻ ✻   4 ✻ ✻ ✻ ✻         4 ✻ ✻ ✻ ✻         1 ✻
b   5   5   4   4   3   3   2  7 ✻ ✻ ✻ ✻ ✻ ✻ ✻ 7 ✻ ✻ ✻ ✻ ✻ ✻ ✻ 6 ✻ ✻ ✻ ✻ ✻ ✻   4 ✻ ✻ ✻ ✻       2 ✻ ✻	c   4   3   3   2   1  5 ✻ ✻ ✻ ✻ ✻ 4 ✻ ✻ ✻ ✻   3 ✻ ✻ ✻     1 ✻	d   1  1 ✻