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Combinatorics / The Research Notebook

Follow The
Argument.

Count a possibility. Challenge a pattern.
Keep the evidence beside the idea.

Exact computationBounded searchUnverified discussion

01 / Enumerate

Change The Constraint.

How much does one forbidden vertex change the count?

Exact integer arithmetic
SE
Admissible paths132

924 without any boundary or blocked vertex.

Recurrence at each vertexD(x,y)=D(x−1,y)+D(x,y−1)D(x,y)=D(x-1,y)+D(x,y-1)

D(0,0) = 1. Forbidden vertices contribute zero. Each last step comes from the left or below.

Tap a vertex to block or restore it. S = (0,0); E = (6,6). Every move is right or up. The displayed route uses R before U in lexicographic order.

Route RRRRRRUUUUUU

From Counting To An Argument

The dynamic count 132 agrees with C(6) = binom(12, 6) / 7 = 132. The formula and dynamic program are two computations; an explanation of why they agree is a separate argument.

NIST: Catalan lattice paths ↗