Search results
Results from the WOW.Com Content Network
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, ... Stanley, Richard P. (2015), Catalan numbers.
Stanley's distinctions include membership in the National Academy of Sciences (elected in 1995), the 2001 Leroy P. Steele Prize for Mathematical Exposition, [7] the 2003 Schock Prize, [3] a plenary lecture at the International Congress of Mathematicians (in Madrid, Spain), [8] and election in 2012 as a fellow of the American Mathematical Society. [9]
The following algebraic identity involving the Catalan numbers = + (), is apparently due to Touchard (according to Richard P. Stanley, who mentions it in his panorama article "Exercises on Catalan and Related Numbers" giving an overwhelming plenitude of different definitions for the Catalan numbers).
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
Whilst the above is a concrete example Catalan numbers, similar problems can be evaluated using Fuss-Catalan formula: Computer Stack : ways of arranging and completing a computer stack of instructions, each time step 1 instruction is processed and p new instructions arrive randomly.
One of those was cult director Richard Stanley, who claimed he saw local deity the White Lady, earning the respect of the close-knit community. As well as being a long-time resident of the area ...
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
Similarly, the Schröder numbers count the number of ways to divide a rectangle into + smaller rectangles using cuts through points given inside the rectangle in general position, each cut intersecting one of the points and dividing only a single rectangle in two (i.e., the number of structurally-different guillotine partitions).