An Extension of Schensted's Theorem

Publisher Summary This chapter presents an extension of Schensted's theorem. σ = ( a 1 , a 2 , …, a n ) is a sequence of integers whose terms are distinct. The chapter presents a theorem that states that the length of the longest increasing subsequence of σ is λ 1 , which is the number of columns of S , and the length of the longest decreasing subsequence is λ 1 *, which is the number of rows of S . According to Schensted's algorithm, if λ = { λ 1 ≥ λ 2 ≥ … ≥ λ q } is a partition of n , a Young tableau of shape λ is a doubly indexed array {s( i, j ) 1 ≤ i ≤ q , 1 ≤ j ≤ λ i } such that the entries s(i, j) are distinct integers, and each row and column forms an increasing sequence.