Algorithms for computing lengths of chains in integral partition lattices

Abstract

Let P l, n denote the partition lattice of l with n parts, ordered by Hardy–Littlewood–Polya majorization. For any two comparable elements x and y of P l, n , we denote by M( x , y ) , m( x , y ) , f( x , y ) , and F( x , y ) , respectively, the sizes of four typical chains between x and y : the longest chain, the shortest chain, the lexicographic chain, and the counter-lexicographic chain. The covers u =(u 1,…,u n)≻ v =(v 1,…,v n) in P l, n are of two types: N-shift (nearby shift) where v i = u i −1, v i+1 = u i+1 +1 for some i; and D-shift (distant shift) where u i −1= v i = v i+1 =⋯= v j = u j +1 for some i and j. An N-shift (a D-shift) is pure if it is not a D-shift (an N-shift). We develop linear algorithms for calculating M( x , y ) , m( x , y ) , f( x , y ) , and F( x , y ) , using the leftmost pure N-shift first search, the rightmost pure D-shift first search, the leftmost N-shift first search, and the rightmost D-shift first search, respectively. Those algorithms have significant applications in complexity analysis of biological sequences.