The large Schröder number Cn counts the number of subdiagonal HVD-lattice paths from (0,0) to (n

The large Schröder number Cn counts the number of...

Key Concepts

Schröder Numbers

Schröder numbers are a sequence of numbers in enumerative combinatorics that count several classes of combinatorial objects. They come in two variants: large Schröder numbers and small Schröder numbers, which count lattice paths and polygon dissections respectively. These numbers appear in problems related to counting lattice paths with multiple allowed step types and in counting the ways to dissect a convex polygon into smaller regions with non-intersecting diagonals.

Subdiagonal HVD-Lattice Paths

Subdiagonal HVD-lattice paths are lattice paths that use horizontal, vertical, and diagonal steps, and are constrained to remain on or below the main diagonal in the grid. This restriction shapes the counting problem and connects the paths with other combinatorial structures such as polygon dissections. The allowed types of moves enable a rich structure in the combinatorial analysis, linking discrete geometry with lattice path enumeration.

Polygon Dissections

Polygon dissections involve dividing a convex polygon into smaller polygonal regions using non-intersecting diagonals. The number of such dissections of a polygon with a given number of sides is closely related to certain combinatorial sequences, such as the small Schröder numbers. This concept is a common theme in studies of planar graphs and geometric combinatorics, illustrating deep connections between geometry and enumerative combinatorics.

Bijection in Combinatorics

A bijection in combinatorics is a one-to-one correspondence between two sets that establishes they have the same cardinality. Establishing a bijection between two combinatorial objects, such as lattice paths and polygon dissections, not only shows that they are counted by the same sequence but also often reveals deeper structural similarities between the problems. Bijections are powerful tools for transferring insights and results from one context to another.

Transcript

00:02 In part a, we're asked to find a formula for the number of vertices of an n pyramid in terms of the number of j faces of an n -minus -1 polytope q.

00:34 So recall that the n -piramid is formed by taking the convex whole of an n -1 polytope q, and a point x which is not in affine space of q so in part a it's pretty clear that f0 of pvn this is the number of vertices we're really only adding one more vertex which is that vertex outside the affine space of q this is going to be f0 of q plus one now in part b we're asked to find the number of k face of an n pyramid where k ranges from 1 all the way up to n minus 2.

01:53 So consider the number of two faces or the number of square faces.

02:01 I guess we should start with number of one faces.

02:03 These are the number of edges of a n pyramid.

02:13 Well, we have all the edges from polytoppe q.

02:20 And then in addition to that, for each one, vertex in the polytopic u i'm going to add exactly one more edge in the n pyramid so it follows that f1 of p vn is equal to f1 of q plus f0 of q now consider the number of square faces of p to n well this is going to be the number of square faces in the polytope q, in addition to that, for each edge in the polytope q, we're going to be adding exactly one more square face in p to the n.

03:23 So the edge in that third point outside the f1 space of q forms a new square face.

03:31 So we have that f2 of p to the n is equal to f1 of q, or sorry, f2 of q plus f1 of q.

03:50 And in fact, you might see a pattern here.

03:53 So you might guess that in general, fk of p to the n is going to be fk of q plus fk minus 1 of q.

04:11 And to see why, well, we have the number of k faces of p to the n.

04:18 This is going to be, we have all the k faces from polytope q.

04:23 And then for each of the k minus 1 faces of q, the addition of the new point in p the n leads to exactly one more k face in pvn.

04:43 So we have one more k face for each k minus one face in q...

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