Livingston and Clark Clement Moore were related by marriage, so it's possible that Livingston wrote the poem and it eventually got to Moore and was published.
A computer analysis said that Livingston was the more likely author.
Henry’s next door neighbor and first cousin was married to a brother-in-law of Moore’s aunt. The story in the Livingston household was that a governess going to work for a family in the south, which could have been the children of his next door neighbor, stopped off at the Moore’s and dropped the poem then.
Do you have any personal knowledge of how this analysis was done? I know of a piece of writing that I have long wished to see put through this same sort of process to determine authorship of it.
The actual name of the poem is “A Visit From St. Nick.”