### Anthony Leverrier

Shared publicly -Anyone who's studied quantum mechanics knows that the subject is largely about pairs of linear operators, a and a⁺, such that:

aa⁺ = a⁺a+1

Solving physics problems often involves rearranging expressions in a and a⁺ so that all of the a⁺ factors are on the left of each monomial and the a factors are on the right. This sort of thing:

aa⁺aa⁺a⁺a⁺aa⁺a

= (a⁺a+1)aa⁺a⁺a⁺aa⁺a

= a⁺aaa⁺a⁺a⁺aa⁺a+aa⁺a⁺a⁺aa⁺a

= ...

= a⁺⁵a⁴+10a⁺⁴a³+23a⁺³a²+9a⁺²a

Getting to the last line takes a substantial amount of work and of course it gets worse when you have infinite sums.

But now I've read http://arxiv.org/abs/0904.1506 I see that there's a much easier way of getting those coefficients: 1, 10, 23, 9.

You can translate a monomial in a and a⁺ into a path on a grid by drawing a⁺ as a horizontal line and a as a vertical line, as in the diagram. That defines a region under the path known as a Ferrers board.

1 is the number of ways of placing zero non-attacking rooks on this Ferrers board. 10 is the number of ways of placing 1 rook, 23 is the number of ways of placing 2 rooks and so on.

I can't believe I've gone all these years without coming across this simple interpretation of the coefficients before.

It's worth reading the proof in the paper. The expression aa⁺ = a⁺a+1 corresponds precisely to a single step in a recursive procedure for counting rook placements.

This is just a tiny hint of the richness of the combinatorics of a and a⁺.

aa⁺ = a⁺a+1

Solving physics problems often involves rearranging expressions in a and a⁺ so that all of the a⁺ factors are on the left of each monomial and the a factors are on the right. This sort of thing:

aa⁺aa⁺a⁺a⁺aa⁺a

= (a⁺a+1)aa⁺a⁺a⁺aa⁺a

= a⁺aaa⁺a⁺a⁺aa⁺a+aa⁺a⁺a⁺aa⁺a

= ...

= a⁺⁵a⁴+10a⁺⁴a³+23a⁺³a²+9a⁺²a

Getting to the last line takes a substantial amount of work and of course it gets worse when you have infinite sums.

But now I've read http://arxiv.org/abs/0904.1506 I see that there's a much easier way of getting those coefficients: 1, 10, 23, 9.

You can translate a monomial in a and a⁺ into a path on a grid by drawing a⁺ as a horizontal line and a as a vertical line, as in the diagram. That defines a region under the path known as a Ferrers board.

1 is the number of ways of placing zero non-attacking rooks on this Ferrers board. 10 is the number of ways of placing 1 rook, 23 is the number of ways of placing 2 rooks and so on.

I can't believe I've gone all these years without coming across this simple interpretation of the coefficients before.

It's worth reading the proof in the paper. The expression aa⁺ = a⁺a+1 corresponds precisely to a single step in a recursive procedure for counting rook placements.

This is just a tiny hint of the richness of the combinatorics of a and a⁺.

4

Small world ;)

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