If *you* are not aware of this, *then* the following, widely known 'interpretations' of the permanent should be pointed out to you.

**0. A definitional interpretation and generalization** (permanents are special *immanants*)**.** Let $\mathbb{L}_n$ denote the lattice of subgroups of the symmetric group $\mathfrak{S}_n$. For each finite permutation group $G\in\mathbb{L}_n$ and any irreducible character $\chi\colon G\rightarrow\mathbb{C}^\times$, the function

$f_\chi^G\colon\color{white}{(a_{i,j})_{(i,j)\in n\times n}}\mathbb{C}^{n\times n}\rightarrow\mathbb{C}$

${\quad}{\quad}\quad\ \ \ (a_{i,j})_{(i,j)\in n\times n}\mapsto \sum_{g\in G}\quad \chi(g)\cdot\prod_{i\in n} a_{i,g(i)}$

is called the *immanant*^{1} w.r.t. $\chi$.

If $\chi$ is the function which is constantly $1$, also called *the principal character*, then the immanant $f_\chi^{\mathfrak{S}_n}$ is just the *permanent* of $(a_{i,j})_{(i,j)\in n\times n}$.

In that sense: *every permanent is an immanant*.

(One can count this as an 'interpretation', sort of. I say 'definitional' above since here the definition of 'permanent' is interpreted in the sense of constructing a larger definition which the permanent is but an instance of.)

**1**.**Conjectural interpretation** (Lieb's Permantal Dominance Conjecture: permanents are top elements of a poset on the set of all irreducible characters of the symmetric group on $n$ elements; not to be confused with the so-called, apparently still *open*)**.**
For each $G\in\mathbb{L}_n$ let $\mathbb{K}_G$ denote the set of all irreducible complex characters of $G$. Then $\mathbb{K}_G$ has the following poset structure $\leq$, in other words, is partially ordered by the following definition.

Let $\chi_0\leq \chi_1$ if and only if (using the notation from **0.**)

for *all* Hermitian positive-semidefinite matrices $A\in\mathbb{C}^{n\times n}$ we have $\frac{f_{\chi_0}^G(A)}{\chi_0(\mathrm{id})}\leq\frac{f_{\chi_1}^G(A)}{\chi_1(\mathrm{id})}$.

It is a famous theorem^{2} of Issai Schur that the *alternating character* $\mathrm{sgn}$ is a bottom element of the poset $(\mathbb{K}_{\mathfrak{S}_n},\leq)$.

(And the immanant w.r.t. $\mathrm{sgn}$ is just the *determinant*.)

A notorious conjecture (and this is what I mean by 'conjectural interpretation' of the permanent) is whether for any $G\in\mathbb{L}_n$ whatsoever, the principal character is a *top element* of the poset $(\mathbb{K}_G,\leq)$.

This goes by the name 'Permanental Dominance Conjecture'.

Explicitly, the (apparently still *open*) Permanental Dominance Conjecture^{3} states:

For every Hermitian positive semidefinite $A\in\mathbb{C}^{n\times n}$

for every subgroup $G$ of $\mathfrak{S}_n$

for every irreducible character $\chi\colon G\rightarrow\mathbb{C}^\times $

we have $\frac{1}{\chi(\mathrm{id})}\sum_{g\in G}\chi(g)\prod_{i\in n}a_{i,g(i)}\leq\mathrm{per}(A)$.

For $n=4$, an explicit Hermitian positive-semidefinite matrix is known which achieves equality in the conjectural equality. (I am writing from memory here; I once saw this matrix, yet do not find it now.)

A 2016 survey on conjectures on inequalities involving permanents is Fuzhen Zhang: An update on a few permanent conjectures. Volume 4, Issue 1 (Aug 2016). From Zhang's work I take the following:

at the end of which we see appear what came to be know by the (to my mind) a little sillily-named and misleadingly-nfamed 'Permanent On Top Conjecture' published by Soules in his 1966 thesis. The conjecture states that the permanent of a Hermitian positive semi-definite matrix $A$ equals the (of necessity real) largest eigenvalue of the *Schur power matrix* of $A$.

Note that Zhang emphasizes something which sounds like that *electronic machines* are necessary to check Shchesnovich's counterexample.

From Zhang's above survey I also take the following 'diagram' (in the informal everyday sense of 'diagram'):

wherein 'Soules POT', a little goofily (I think^{4}), refers to the 'Permanent On Top' conjecture, and wherein 'Lieb per-dom' refers to the open 'Permanental Dominance Conjecture', and wherein I colored red what seems to now be known to be semantically false, colored yellow what seems to be open conjectures, and colored green the material/syntactic implications which are known to exist. (Soules, Bapat and Sunder can take solace in the fact that there are many mathematicians who find material implications at least as interesting as semantic truth-values; also, from what I understand of this so far, the *certification of Shchesnovich's disproof seems to be an interesting and useful problem in itself*.^{5})

**EDIT:** As an exercise in the important method of criticizing, and then *improving* one's *own* answers, let me add the following

In light of Ryo Taba's publication, it seems *wrong* to say that the permanent can be interpreted as the top element of a poset structure. Of a good interpretation I demand that it is right through and through (as is e.g. the interpretation of the rhythm of day and night as the effect of a tilted axis of revolution), which *this* interpretation just does not seem to be: it is too weak a statement in dimension $n=3$, as Taba's sharper estimate shows.

^{1} _{Language-hazard: the correct spellings in contemporary mathematical discourse are like given here: imman$\mathbf{a}$nt, yet perman$\mathbf{e}$nt. Note that 'imman$\mathbf{e}$nt' lexically also exists. Note also that there is the following randomness: in the case of 'permanent' the usual adjective was also made the noun. In the case of 'immanent', for some reason people avoided making the adjective the noun, and used the variant spelling 'immanant'. }

^{2} _{ Issai Schur: Uber endliche Gruppen und Hermitische Formen, Mathematische Zeitschrift 1, p. 184-207.
}

^{3} _{ It was pointed out to me recently (I had not kept up with this, was partly writing from what I remembered from what I learned when I worked on counting sign-matrices, and was unaware of this 2015-development when I wrote my answer) that a disproof of 'Soules POT' has recently been published: Valery S. Shchesnovich: The permanent-on-top conjecture is false. Linear Algebra and its Applications
Volume 490, 1 February 2016, Pages 196-201 The counterexample is given in dimension $n=5$. Also, there is a stunningly elementary proof by Ryo Tabata in Hiroshima Mathematical Journal 40 (2010), 205-213 that the Permanental Dominance Conjecture is 'more than true' in dimension $n=3$ and $\large \text{for $G=$alternating group on $n$ elements}$: discounting the introduction and references, Ryo Tabata on a mere five pages gives a proof, readable by
anyone knowing high-school calculus and linear algebra over $\mathbb{C}$, of a weird strengthening of the Permantental Domimance Conjecture, replacing the upper bound $\mathrm{per}(A)$ by $\lambda\cdot \mathrm{per}(A) + (1-\lambda)\cdot \mathrm{det}(A)$ where $\lambda=\frac{1}{\sqrt[3]{2}}$. I.e., a nontrivial convex linear combination of the permanent and the determinant is an upper bound in dimension $n=3$. $\large \text{Again, please note that Tabata's proof was formulated for characters of the alternating group only.}$ }

In dimension $n=4$, even the truth of 'Soules POT' seems to be an open problem. (Cf. remarks of Zhang in loc. cit. to the effect that Shchesnovich *tried* to disprove it)

^{4} _{ I think it is fair to say that the publisher of the survey I cite should have given Zhang better editorial advice and services. One linguistic infelicity chases the next. It is not the author who is to blame. }

^{5} _{ This is not to cast doubt on the correctness of Shchesnovich's result. I was led to write this because it seems that no one has not used numerical computations when checking Shchesnovich's result. And numerical computations are an important science of its own, full of pitfalls. }