Answer: E

Solution:

Sort the five digits as $d_{(1)}\le d_{(2)}\le\cdots\le d_{(5)}.$ The requirement "at least $j$ digits less than $j$" is equivalent to $d_{(j)}\le j-1$ for $j=1,2,3,4,$ i.e. $d_{(1)}=0,\ d_{(2)}\le1,$ $\ d_{(3)}\le2,\ d_{(4)}\le3$ (with $d_{(5)}\le4$ automatic).

These strings are exactly the parking functions of length $5.$ To count them, arrange $6$ parking spaces in a circle and let $5$ labeled cars choose arbitrary preferred spaces. Each car moves forward to the first open space. Among the $6^5$ preference strings, rotating all preferences cycles the unique empty space through all $6$ positions. Therefore exactly $6^5/6=6^4=1296$ strings leave a specified space empty. Choosing that space as the extra sixth space gives precisely the sorted inequalities above.

Thus, the correct answer is E.