Is there a name for the infinite sequence of k-multisets with one element, two elements, etc.? Say you write down all $k$-multisets with elements taken from {0}. (Obviously there's just one, which is $k$ repetitions of "0".) Then list all $k$-multisets taken from {0,1} that haven't been listed, in lexicographical order. Then list $k$-multisets taken from {0,1,2} not yet listed, {0,1,2,3}, and so on. In other words, all $k$-multisets are listed with new elements added "one at a time." Is there a name for this sequence? It seems like it would be useful, such as for finding sequences of $k$ natural numbers with some property, but I haven't found the sequence referred to explicitly.
Example with $k=3$:
000
001
011
111
002
012
112
022
122
222
003
013
113
023
123
223
033
133
233
333
...
Alternately: in the previous list, if you replace each multiset with the list of permutations of that multiset in lexicographical order, is there a name for the resulting list of permutations?
 A: I found a partial answer. In The Art of Computer Programming 7.2.1.3, Knuth calls this generating multicombinations. In fact, the $d_0d_1d_2$ column of Table 1 is exactly my example but with the strings reversed! Knuth's Algorithms L or T (T is a faster version of L) enumerates all combinations in lexicographical order (what my list is with the strings reversed). Actually, it generates all combinations of $n+k-1$ elements taken $k$ at a time without replacement, which is equivalent to my list of $k$-combinations of $n$ elements with replacement, and many other equivalent representations. It's not too hard to adapt Algorithm L or T to enumerate multicombinations.
That seems to settle combinations, but I haven't found a name for the sequence of permutations I mentioned, generated by:


*

*for each $n$ = 0,1,2,...


*

*for each multicombination $c$ of natural numbers up to $n$ taken $k$ at a time in lexicographical order:


*

*for each permutation $p$ of $c$ in lexicographical order:


*

*visit $p$




