  Karonski, Luczak, and Thomason (2004) conjectured that, for any connected
graph G on at least three vertices, there exists an edge weighting from {1,2,3}
such that adjacent vertices receive different sums of incident edge weights.
Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each
edge's weight may be chosen from an arbitrary list of size 3 rather than
{1,2,3}. We examine a variation of these conjectures, where each vertex is
coloured with a sequence of edge weights. Such a colouring relies on an
ordering of the graph's edges, and so two variations arise -- one where we may
choose any ordering of the edges and one where the ordering is fixed. In the
former case, we bound the list size required for any graph. In the latter, we
obtain a bound on list sizes for graphs with sufficiently large minimum degree.
We also extend our methods to a list variation of irregularity strength, where
each vertex receives a distinct sequence of edge weights.
