  By introducing $k$-marked Durfee symbols, Andrews found a combinatorial
interpretation of $2k$-th symmetrized moment $\eta_{2k}(n)$ of ranks of
partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment
$\mu_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order
spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation
of $\mu_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a
representation of ordinary partitions given by Dyson, and we show that
$\mu_{2k}(n)$ equals the number of $(k+1)$-marked Dyson symbols of $n$. We then
introduce the full crank of a $k$-marked Dyson symbol and show that there exist
an infinite family of congruences for the full crank function of $k$-marked
Dyson symbols which implies that for fixed prime $p\geq 5$ and positive
integers $r$ and $k\leq (p-1)/2$, there exist infinitely many non-nested
arithmetic progressions $An+B$ such that $\mu_{2k}(An+B)\equiv 0\pmod{p^r}$.

Combinatorics;Number Theory