An asymptotic version of a theorem of Knuth

A well-known theorem of Knuth establishes a bijection between permutations in S(N) with no decreasing subsequence of length three and rectangular standard Young tableaux of shape R(2,N). We prove an asymptotic version of this result: for any fixed integer d⩾2, the number of permutations in S(dn) with no decreasing subsequence of length d+1 is asymptotically equal, as n→∞, to the number of standard Young tableaux on the rectangle R(d,2n). This yields a new proof of Regevʼs theorem on the asymptotic number of permutations without long decreasing subsequences, and consequently an alternative, elementary evaluation of Mehtaʼs integral at β=2.