The markings indicated:
"56. A. 110. NYV”
most logically reads as:
56 — rack or weapon number
A — Company A
110 — 110th Regiment
NYV — New York Volunteers
The actual August 25, 1862 muster-in roll for Company A, 110th New York State Volunteers has a column headed "Number each person.” On the Company A continuation page, the privates are numbered sequentially. NY Military & Naval Affairs
And the entry at No. 56 is:
56 — David Perry, Private, Company A, 110th New York State Volunteers. NY Military & Naval Affairs
56 — exactly the number assigned to David Perry on the Company A muster-in roll
A — Company A
110 — 110th Regiment
NYV — New York Volunteers
It should be kept in mind that this information does not exclusively confirm that David Perry actually carried this rifle, however, it does strongly suggest the issue of this weapon to that soldier.