This is really two questions in one. First I need a proof or disproof of the conjectured switchboard/diner model conjectured below. Second I would like to know of other models.

The unsigned, Chebyshev, or probabilist’s, Hermite polynomials $H_n(x)$ (OEIS A099174, Wikipedia, MathWorld, see bottom for the first few) have the exponential generating function

$e^{t^2/2} ; e^{tx} = e^{th.} ; e^{tx} = e^{t(h.+x)} = e^{tH.(x)} = sum_{n geq 0} H_n(x) ; frac{t^n}{n!} $

where $e^{t^2/2} = e^{h.t}$ with $h_n$ the aerated, odd double factorials OEIS A001147

and

$H_n(x) = (h.+x)^n = sum_{k=0}^n ; binom{n}{k} ; h_{k} ; x^{n-k}.$

They are classic Sheffer Appell polynomials (two other Appell sequences are the Bernoulli and the fundamental powers $p_n(x) = x^n$) with the raising op $R_H = x + d/dx = x + D$ and. as for all Appell sequences, the lowering op $L_H = D$; that is,

$R_H ; H_{n+1}(x) = (x+D) ; H_n(x) = H_{n+1}(x)$

and

$L_H ; H_n = D ; H_n(x) = D ; (h.+x)^n = n ; H_{n-1}(x).$

Consequently,

$R_H^n 1 = (x +D)^n 1 = H_n(x),$

and it turns out (see OEIS A344678) that the normal ordering of $(x+D)^n$, i.e., ordering all derivatives to the right of any $x$ via the Leibniz Lie commutator $(D,x) = Dx – xD = 1$, gives an expression equivalent to $H_n(x+y)$ with all $y$‘s to the right. For example, the noncommutative operations

$(x+D)^2 = xx + xD + Dx + DD = x^2 +xD + xD +1 + D^2 = x^2 + 2xD + 1 + D^2$

give the same result as the commutative operations

$H_2(x+y) = (h.+x+y)^2 = (x +H.(y))^2 = x^2 + 2xH_1(y) + H_2(y) = x^2 + 2xy + 1 + y^2$

or

$H_2(x+y) = (x+y)^2 + 1 = x^2 + 2xy + 1 + y^2.$

I can prove this general equivalence in results of the commutative calculation of the $H_n(x+y)$ polynomials and the normal ordering of the $2^n$ permutations, $(x+D)^n$, of the symbols $x$ and $D$ subject to the Leibniz commutator relation $(D,x) = 1$ of the Heisenberg-Weyl algebra.

This naturally generalizes to the ladder ops–the lowering/destruction/annihilation, $L$, and raising/creation, $R$, ops–of any Sheffer polynomial sequence, giving the equivalence in form of the monomial rep of $H_n(x+y)$ and the normal ordering of $(L+R)^n$.

Now for the combinatorial models:

The Donaghey ref in OEIS A005425 gives $H_n(2)$ as the number of ways $n$ subscribers to a switchboard could be talking either to another subscriber, someone else on an outside line, or not at all–no conference calls allowed; i.e., a person can be talking at most with one other person. This can be viewed as a dinner scenario with each diner among $n$ diners either exchanging seats with another diner; remaining seated; or getting up, changing plans, and sitting back down–at most only one exchange per person allowed.

Apparently, from examining the coefficients of the distinct monomials of the first four $H_n(x+y)$, the coefficients give a finer tabulation of these exchanges. The first five are

$H_0(x+y) = 1,$

$H_1(x+y) = x + y,$

$H_2(x+y) = x^2 + 2 x y + 1 + y^2,$

$H_3(x) = x^3 + 3 x^2 y + 3 x + 3 x y^2 + 3 y + y^3,$

$H_4(x+y) = x^4 + 4 x^2 y + 6 x^2 + 6 x^2 y^2 + 12 x y + 4 x y^3 + 3 + 6 y^2 + y^4,$

$H_5(x+y) = x^5 + 5 x^4 y + 10 x^3 + 10 x^3 y^2 + 30 x^2 y + 10 x^2 y^3 + 15 x + 30 x y^2 + 5 x y^4 + 15 y + 10 y^3 + y^5.$

Examples of the relation to the switchboard scenario:

$H_1(x+y)$ correspond to one subscriber either offline or talking to a non-subscriber via an outside line.

The coefficient $12$ in $H_4(x+y)$ corresponds to the number of ways that, among $4$ subscribers, one pair is in mutual conversation, another subscriber is on an outside line, and the remaining subscriber is offline. The $3$ in the polynomial corresponds to the number of ways two pairs among the four subscribers could be talking.

The Hermite polynomials and their relationships to diverse combinatorial and analytic scenarios have been fairly thoroughly researched and much has been written on the various families of Hermite polynomials, so though there is likely one in the literature, it’s rather hard to find a proof of the above conjecture. Can anyone provide a proof or a link to one?

Some analytics and refs that might prove useful:

Again $H_n(2)$ is given by A005425 = 1, 2, 5, 14, 43, 142, 499, 1850 … .

Number of monomials for each polynomial is 1, 2, 4, 6, 9, 12, 16, 20, 25 … A002620(n).

$h_n$ are the aerated odd double factorials A001147 1, 0, 1, 0, 3, 0, 15, 0, 105, …

$H_n(x+y) = (H.(x)+y)^n = (h.+x+y)^n$.

The coefficient of $x^k y^m$ in $H_n(x+y)$ is $frac{n!}{(n-k-m)! ; k! ; m!} ; h_{n-k-m} $.

The first few Hermite polynomials are (unsigned A099174)

$H_0(x) = 1,$

$H_1(x) = x,$

$H_2(x) = x^2 + 1,$

$H_3(x) = x^3 + 3x,$

$H_4(x)= x^4 + 6 x^2 + 3,$

$H_5(x) = x^5 + 10x^3 + 15x,$

$H_6(x) = x^6 + 15x^4 + 45x^2 + 15.$

“Combinatorial Models of Creation-Annihilation” by Blasiak and Flajolet.