Random Young Tableaux

We derive new combinatorial identities which may be viewed as multivariate analogs of summation formulas for hypergeometric series. As in the previous paper Re], we start with probability distributions on the space of the innnite Young tableaux. Then we calculate the probability that the entry of a random tableau at a given box equals n = 1; 2; : : :. Summing these probabilities over n and equating the result to 1 we get a nontrivial identity. Our choice for the initial distributions is motivated by the recent work on harmonic analysis on the innnite symmetric group and related topics. Let Tab be the set of all innnite standard Young tableaux T = (T(i; j)). Given a probability measure M on Tab, we may speak about the random innnite tableau T. Let P M (T(i; j) = n) denote the probability that T has the entry n at the box (i; j). Fix a box (i; j) such that the shape of T contains (i; j) almost surely. Then X n0 P M (T(i; j) = n) = 1: (0.1) In Re], it was shown that by specializing M one can get from (0.1) many nontriv-ial identities. These identities look as summation formulas for multivariate series of hypergeometric type. For instance, one of the identities is as follows (see Re, (4.2.1 0)]): X p 1 >>p k 1 q 1 >>q l 1 (jpj + jqj + 1 2 k + l ? (k + l) 2 ])! V 2 (p) V 2 (q) Q 1rk (p r !) 2 Q 1sl (q s !) 2 Q 1rk Q 1sl The identity (0.2) arises from the so{called Plancherel measure, the xed box being (k + 1; l + 1). From a diierent point of view, the identity (0.2) (written in an equivalent form) is discussed in MMW]. 1 In the present paper, which is a continuation of Re], we derive new identities of the form (0.1). The results are as follows. t k+r 1 ++r l +1 (t) s 1 ++s k +r 1 +2r 2 ++lr l +kl+k+l+1 = 1; (0.3) where (x) n = x(x + 1) : : :(x + n ? 1) is the Pochhammer symbol. See Theorem 3.3.2 below. We present two proofs of (0.3). One of them follows our general scheme while another is a direct argument, which is largely due to S. Milne …