Most mathematicians are aware that our species consists of two genders, denoted for simplicity by the multisets $\lbrace X,X\rbrace$ and $\lbrace X,Y\rbrace$, with offspring given by $\lbrace A,B\rbrace$ for $A\in\lbrace X,X\rbrace$ and $B\in \lbrace X,Y\rbrace$.

I am asking for the existence of combinatorial structures generalizing this construction.

More precisely, given a finite set $\mathcal C$ of $n$ distinct elements ($n=2$ and $\mathcal C=\lbrace X,Y\rbrace$ in the above example) and an integer $k$, we denote by $\mathcal M_k$ the set of all multisets containing exactly $k$ not necessarily distinct elements of $\mathcal C$.

A *gender partition* of a subset $\mathcal S\subset \mathcal M_k$ is a partition of $\mathcal S$ into $k$ non-empty parts $\mathcal G_1,\dots,\mathcal G_k$ called genders such that the following two conditions hold:

(i) Given $(g_1,\dots,g_k)\in\mathcal G_1\times \dots \times \mathcal G_k$, every element $(x_1,\dots,x_k)\in g_1\times \dots\times g_k$ gives rise to a multiset $\lbrace x_1,\dots,x_k\rbrace$ which is in $\mathcal S$.

(ii) Every multiset $g\in\mathcal S$ is of the form $\lbrace x_1,\dots,x_k\rbrace$ for $(x_1,\dots,x_k)\in g_1\times \dots\times g_k$ where $g_i\in \mathcal G_i$ are suitable elements.

Examples with $\mathcal C=\lbrace X,Y\rbrace$ are:

(a) $\mathcal S= \lbrace X,X\rbrace\cup \lbrace X,Y\rbrace$ with $\mathcal G_i$ given by singletons.

(b) $\mathcal S= \lbrace \lbrace X,X\rbrace,\lbrace Y,Y\rbrace\rbrace \cup \lbrace X,Y\rbrace$ with $\mathcal G_1=\lbrace \lbrace X,X\rbrace,\lbrace Y,Y\rbrace\rbrace$ and $\mathcal G_2$ consisting of $\lbrace X,Y\rbrace$.

(c) An example with $k=3$ (easily generalizable to arbitrary values of $k$) is given by $\mathcal S=\lbrace \lbrace X,X,X\rbrace,\lbrace X,X,Y\rbrace, \lbrace X,Y,Y\rbrace\rbrace$ with $\mathcal G_i$ given by singletons.

More examples of gender partitions $\mathcal S=\mathcal G_1\cup \dots\cup \mathcal G_k$ are fairly easy to construct. (And there are fairly easy notions for "products", "quotients", one can split an element of $\mathcal C$ into several new elements, etc.)

The following additional condition is more difficult to satisfy:

Call a gender partition $\mathcal S=\mathcal G_1\cup \dots\cup\mathcal G_k$
*balanced* if $\mathcal S$ admits a stationary probability measure
$\mu$ giving equal weight $\frac{1}{k}$ to all genders $\mathcal G_i$.
A probability measure $\mu$ on $\mathcal S$ is *stationary* if
the probabiliy $\mu(\lbrace x_i,\dots,x_k\rbrace)$ of every offspring
of $(g_1,\dots,g_k)$ (with respect to uniform choices for $x_i\in g_i$)
is proportional to $\prod_{i=1}^k\mu(g_i)$. (Stationary probability
measures exist always and are unique if $\mathcal S$ is
minimal in some sense.)

Example: The examples (a) and (b) above are balanced, (c) is not balanced.

Question: Produce other examples of balanced gender partitions. Is there for example a balanced gender partition for $k=3$?

Remark: One can also consider probabilities on offsprings which depend on the choice of $x_i\in g_i$. Example (c) is not balanced even in this more general framework.

Variation: Instead of working with multisets, one can also work with sequences of length $k$. An offspring of $k$ sequences $g_1=(g_1(1),\dots,g_1(k)),\dots,g_k=(g_k(1),\dots,g_k(k))$ with $g_i\in\mathcal G_i$ is then given by $g_1(\sigma(1)),\dots,g_k(\sigma(k))$ where $\sigma$ is a (not necessarily arbitrary) permutation of $\lbrace 1,\dots,k\rbrace$.

finiteset of genders, and I've heard that the mating rituals there are quite a party. $\endgroup$ – Joel David Hamkins May 11 '10 at 16:14