Thirty four comparisons are required to sort 13 items

Posets with seven linear extensions sortable by three comparisons

“…comparisons are necessary (see Knuth [5]). However, better lower bounds are obtained: Wells [8] obtained the greater lower bound for 12 elements, Kasai et al [4] for 13 elements, and Peczarski [6,7] for 14, 15, and 22 elements, by exhaustive computations. Ford-Johnson algorithm [2] gives upper bounds for the minimum number of comparisons.…”

Towards Optimal Sorting of 16 Elements

“…Carrying an exhaustive computer search, Wells discovered in 1965 [9,10] that the FJA is optimal for 12 elements and S(12) = F (12) = C(12)+1 = 30. Kasai et al [3] computed S(13) = F (13) = C(13) + 1 = 34 in 1994, but that result was not widely known. It was discovered again a few years later [5], independently, extending the Wells method.…”