Enumeration of Semi { Standard Young Tableaux

Standard tableaux. A Young tableau (also known as a standard tableau) is an arrangement of distinct integers in an array of left justiied rows, such that the entries of each row are in increasing order from left to right and the entries of each column are in increasing order from bottom to top 2. For instance, here is an array of size 15: 11 4 8 12 14 3 6 7 13 1 2 5 9 10 15 : In general a tableau of n cells has a shape described by a partition of the integer n, the shape of the example being (6; 4; 4; 1). Such tableaux originate in the study of linear representations of the symmetric group. The main result is the following: Each permutation of 1::n] is uniquely representable by a pair of tableaux of size n having the same shape. The eeective mapping|based on the \bumping rule"|constitutes the celebrated Robinson{Schensted correspondence. A very readable introduction to the subject is to be found in Knuth's book 4, Sec. 5.1.4]. Many properties result from the Robinson{Schensted correspondence. First, the total number of tableaux of size n is equal to the number of involutions (these are permutations such that 2 = Id) of 1::n], that is T n = n! z n ] exp(z + z 2 2): Next, the length of the longest increasing subsequence of a permutation coincides with the common base size of the pair of tableaux associated with. The enumeration of tableaux of bounded width or height thus appears of interest in connection with various order statistics, these structures being also rich combinatorial objects per se. Let T k] n be the number of tableaux of height k, and denote by C n = 1=(n + 1)