### Dan Piponi

Shared publicly -A plugboard (more specifically an n-plugboard) is a board with n holes in it, each of which can accept the end of a wire. Pick one representative plugboard for each n and call it P(n). We can make a vector space over k, say, (which I’ll call PB) whose basis elements are {P(0), P(1), P(2), P(3), …}.

We can define a product ⊗ on this vector space and make it a commutative algebra. Given two plugboards we can stick them together and then plug in some wires with one end in each of the original parts. Each wire used results in one less hole in each of the two parts. For example, after joining an m-plugboard to an n-plugboard and plugging in k wires we have, in effect, an (m+n-k)-plugboard left over. Define the product of P(m) and P(n) in PB as the sum a(i)P(i) where a(i) is the number of ways you can get an i-plugboard by adding wires from a P(m) to a P(n) joined to it. We don’t care about the order in which the wires are added, which way round the wires are or anything else. One wire is the same as any other. We’re only counting the choices of holes they connect. Extend this multiplication to all of PB by linearity. P(0)=1, the unit.

For example P(2)⊗P(3) = P(5)+6P(3)+6P(1). There is one way of adding no wires from a P(2) to a P(3). There are 3+3 ways to add one wire going from a P(2) to a P(3) leaving a P(3). And there are 3x2 ways to add two wires going from a P(2) to a P(3) leaving a P(1). More generally, a ⊗-product of n plugboards enumerates the ways you can hook up any number of wires with each wire going between two different plugboards.

The Enigma machine from World War II had a plugboard pictured above with 26 holes in it and 10 wires. The number of ways those wires could be plugged in is given by the coefficient of P(6) in P(1)⊗P(1)⊗…26 factors in total…⊗P(1).

So here’s a neat theorem I’m not going to prove: there is an isomorphism ϕ from (PB,+,⊗,0,1) to the algebra of polynomials over k (k[x],+,×,0,1) given by PB(n) → H(n,x) where H(n,x) is the nth “probabilist’s” Hermite polynomial [3]. The ⊗-product becomes ordinary multiplication.

For example H(2,x)H(3,x) = H(5,x)+6H(3,x)+6H(1,x). The number of ways to use the Enigma plugboard is the coefficient of H(6,x) when the polynomial x^26 is written as a sum of Hermite polynomials. (Which is also the coeffcient of x^6 in H(26,x) but that’s another story.)

The theorem is essentially Isserlis’ theorem in disguise, aka Wick’s theorem. It forms the cornerstone of quantum field theory and it's what leads to Feynman diagrams. But I like looking at it in terms of plugboards. (Compare also the connection to wiring up modular synths I mentioned a while back [5].)

There’s another operation you can perform on an n-plugboard - you could just block one hole giving an (n-1)-plugboard. If a plugboard has n holes there are n ways you could do this. So if we want to count the ways to block up holes it makes sense to consider the operator ∂:PB → PB such that ∂(P(n)) = nP(n-1). It’s not too hard to convince yourself that ∂ satisfies the Leibniz rule ∂(ab)=a∂b+(∂a)b. When you block a hole in a product you either block a hole in the first component or block a hole in the second. So we have a differential algebra [2].

The map ϕ is actually an isomorphism of differential algebras and ∂ maps to the ordinary derivative under the isomorphism. Many theorems about derivatives of polynomials carry over to plugboards and vice versa. For example, arguing informally now so as to allow infinite sums (or moving to a suitable completion of PB), think about e = P(0)/0!+P(1)/1!+PB(2)/2!+P(3)/3!+… . This satisfies the “differential equation” ∂e = e. So by the isomorphism we must have ∂ϕ(e) = ϕ(e). But ϕ(e) is essentially a formal power series in x. So solving the “differential equation” we get ϕ(e) = Aexp(x) for some constant A. In other words H(0,x)/0!+H(1,x)/1!+H(2,x)/2!+H(3,x)/3!+ … = Aexp(x) for some A. (A is exp(-1/2) BTW. That exp(-1/2) factor is a reminder that we’re not talking about complete trivialities here.) We can "transfer" other familiar functions like sin and cos from the reals to PB.

There’s lots more I could say. PB tells you all about functions of Gaussian random variables. (I was hoping to use this for some statistics computations but it isn't going to work out...) The isomorphism ϕ is, in some sense, a Gaussian blur(!). There is a shadowy conspiracy theory going on behind all this stuff [6]. But this is enough for now.

[1] https://en.wikipedia.org/wiki/Isserlis%27_theorem

[2] https://en.wikipedia.org/wiki/Differential_algebra

[3] https://en.wikipedia.org/wiki/Hermite_polynomials

[4] https://en.wikipedia.org/wiki/Enigma_machine#/media/File:Enigma-plugboard.jpg

[5] https://plus.sandbox.google.com/+DanPiponi/posts/DJuDKHeGxfy

[6] https://en.wikipedia.org/wiki/Umbral_calculus

We can define a product ⊗ on this vector space and make it a commutative algebra. Given two plugboards we can stick them together and then plug in some wires with one end in each of the original parts. Each wire used results in one less hole in each of the two parts. For example, after joining an m-plugboard to an n-plugboard and plugging in k wires we have, in effect, an (m+n-k)-plugboard left over. Define the product of P(m) and P(n) in PB as the sum a(i)P(i) where a(i) is the number of ways you can get an i-plugboard by adding wires from a P(m) to a P(n) joined to it. We don’t care about the order in which the wires are added, which way round the wires are or anything else. One wire is the same as any other. We’re only counting the choices of holes they connect. Extend this multiplication to all of PB by linearity. P(0)=1, the unit.

For example P(2)⊗P(3) = P(5)+6P(3)+6P(1). There is one way of adding no wires from a P(2) to a P(3). There are 3+3 ways to add one wire going from a P(2) to a P(3) leaving a P(3). And there are 3x2 ways to add two wires going from a P(2) to a P(3) leaving a P(1). More generally, a ⊗-product of n plugboards enumerates the ways you can hook up any number of wires with each wire going between two different plugboards.

The Enigma machine from World War II had a plugboard pictured above with 26 holes in it and 10 wires. The number of ways those wires could be plugged in is given by the coefficient of P(6) in P(1)⊗P(1)⊗…26 factors in total…⊗P(1).

So here’s a neat theorem I’m not going to prove: there is an isomorphism ϕ from (PB,+,⊗,0,1) to the algebra of polynomials over k (k[x],+,×,0,1) given by PB(n) → H(n,x) where H(n,x) is the nth “probabilist’s” Hermite polynomial [3]. The ⊗-product becomes ordinary multiplication.

For example H(2,x)H(3,x) = H(5,x)+6H(3,x)+6H(1,x). The number of ways to use the Enigma plugboard is the coefficient of H(6,x) when the polynomial x^26 is written as a sum of Hermite polynomials. (Which is also the coeffcient of x^6 in H(26,x) but that’s another story.)

The theorem is essentially Isserlis’ theorem in disguise, aka Wick’s theorem. It forms the cornerstone of quantum field theory and it's what leads to Feynman diagrams. But I like looking at it in terms of plugboards. (Compare also the connection to wiring up modular synths I mentioned a while back [5].)

There’s another operation you can perform on an n-plugboard - you could just block one hole giving an (n-1)-plugboard. If a plugboard has n holes there are n ways you could do this. So if we want to count the ways to block up holes it makes sense to consider the operator ∂:PB → PB such that ∂(P(n)) = nP(n-1). It’s not too hard to convince yourself that ∂ satisfies the Leibniz rule ∂(ab)=a∂b+(∂a)b. When you block a hole in a product you either block a hole in the first component or block a hole in the second. So we have a differential algebra [2].

The map ϕ is actually an isomorphism of differential algebras and ∂ maps to the ordinary derivative under the isomorphism. Many theorems about derivatives of polynomials carry over to plugboards and vice versa. For example, arguing informally now so as to allow infinite sums (or moving to a suitable completion of PB), think about e = P(0)/0!+P(1)/1!+PB(2)/2!+P(3)/3!+… . This satisfies the “differential equation” ∂e = e. So by the isomorphism we must have ∂ϕ(e) = ϕ(e). But ϕ(e) is essentially a formal power series in x. So solving the “differential equation” we get ϕ(e) = Aexp(x) for some constant A. In other words H(0,x)/0!+H(1,x)/1!+H(2,x)/2!+H(3,x)/3!+ … = Aexp(x) for some A. (A is exp(-1/2) BTW. That exp(-1/2) factor is a reminder that we’re not talking about complete trivialities here.) We can "transfer" other familiar functions like sin and cos from the reals to PB.

There’s lots more I could say. PB tells you all about functions of Gaussian random variables. (I was hoping to use this for some statistics computations but it isn't going to work out...) The isomorphism ϕ is, in some sense, a Gaussian blur(!). There is a shadowy conspiracy theory going on behind all this stuff [6]. But this is enough for now.

[1] https://en.wikipedia.org/wiki/Isserlis%27_theorem

[2] https://en.wikipedia.org/wiki/Differential_algebra

[3] https://en.wikipedia.org/wiki/Hermite_polynomials

[4] https://en.wikipedia.org/wiki/Enigma_machine#/media/File:Enigma-plugboard.jpg

[5] https://plus.sandbox.google.com/+DanPiponi/posts/DJuDKHeGxfy

[6] https://en.wikipedia.org/wiki/Umbral_calculus

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Another interesting connection: an example of a plugboard in nature is the way the bases in RNA can bind to each other. So if you enumerate all the ways a sequence of n bases can bind to itself you're led inevitably to Hermite polynomials. To see what I'm talking about peek here http://ipht.cea.fr/Docspht/articles/t04/135/publi.pdf

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