9873

Lattice Path Interpretations

Many famous families of integers can be represented by the number of paths through a lattice given various restrictions. This Demonstration illustrates the various paths from to using northeast and southeast steps, with the option to permit no eastward steps, single steps east, or double steps east, as well as the option to accept paths dipping below the axis (southeast steps). Counting the paths for the various choices generates six named sequences.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: paths consisting only of steps northeast and southeast that are allowed to drop below the axis represent the central binomial coefficients—that is, the middle entries in Pascal's triangle
Snapshot 2: paths that do not cross the axis and consist of single steps northeast and southeast and double steps east describe the Schröder numbers
Snapshot 3: paths that can cross the axis and consist only of single steps northeast, southeast, and east are equivalent to the sums of squares of the trinomial coefficients—that is, the coefficients in the expansion of
Identification of "sums of squares of trinomial coefficients" (A082758) thanks to the On-Line Encyclopedia of Integer Sequences: oeis.org.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+