The $1$s on the edges of Pascal's triangle appear infinitely many times in the array.
One number ‒ just one ‒ appears exactly once.
Infinitely many have multiplicity $2.$
Infinitely many have multiplicity $3.$
Infinitely many have multiplicity $4.$
Infinitely many have multiplicity $6.$
One is known to have multiplicity $8;$ whether any others do is unknown.
Whether any odd multiplicities more than $3$ occur is unknown.
Singmaster's conjecture is that the set of multiplicities is bounded, so it may be that no multiplicities exceeding $8$ occur.
(The one known to have multiplicity $8$ is $\dbinom{3003} 1 = \dbinom{78} 2 = \dbinom{15} 5 = \dbinom{14} 6.$)
- The multiplicities known to occur infinitely many times are $2,3,4,6.$
- The only rotational symmetries of a two-dimensional lattice are of order $2,$ $3,$ $4,$ or $6.$
Might there be some connection between the two problems that explains why that same set of numbers appears? (Here's a wild fantasy: Prove the statement about Pascal's triangle by using the statement about symmetries of lattices.)