the number of double permutations (sequences of the numbers from 1 to n, each number appearing twice, with the first occurrences of each number in sorted order) that avoid the permutation patterns 12312 and 121323. Related sequences The closely related large Schröder numbers are equal to twice the Schröder–Hipparchus numbers, and may also be used to count several types of combinatorial objects including certain kinds of lattice paths, partitions of a rectangle into smaller rectangles by recursive slicing, and parenthesizations in which a pair of parentheses surrounding the whole sequence of elements is also allowed. The Catalan numbers also count closely related sets of objects including subdivisions of a polygon into triangles, plane trees in which all internal nodes have exactly two children, and parenthesizations in which each pair of parentheses surrounds exactly two symbols or parenthesized groups. The sequence of Catalan numbers and the sequence of Schröder–Hipparchus numbers, viewed as infinite-dimensional vectors, are the unique eigenvectors for the first two in a sequence of naturally defined linear operators on number sequences. More generally, the kth sequence in this sequence of integer sequences is (x1, x2, x3, ...) where the numbers xn are calculated as the sums of Narayana numbers multiplied by powers of k. This can be expressed as a hypergeometric function: Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. 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. The corresponding numbers for the permutohedron are respectively the ordered Bell numbers and the factorials. Recurrence As well as the summation formula above, the Schröder–Hipparchus numbers may be defined by a recurrence relation: Stanley proves this fact using generating functions while Foata and Zeilberger provide a direct combinatorial proof. History Plutarch's dialogue Table Talk contains the line: Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.) This statement went unexplained until 1994, when David Hough, a graduate student at George Washington University, observed that there are 103049 ways of inserting parentheses into a sequence of ten items. The historian of mathematics Fabio