Generalizations of Pascal’s Triangle
Robert Ronan, Vassar College ’15, Anastasia Stevens, Vassar College ’15, Lilian Zhao, Vassar College ’17 and Prof. John McCleary
Pascal’s triangle is the triangular array of binomial coefficients. The coefficients in the nth row (starting with row n=0) are `n choose k', numbered from left to right beginning with k=0. To construct each row, k=0 always has the value 1, and subsequent coefficients are determined by the sum of the two coefficients directly above. A binomial coefficient is also the coefficient of the x^k term in the polynomial expansion of the binomial power (1+x)^k, or, equivalently, the number of ways to select k objects from a set of n distinct things.
Our research began with a focus on various properties of binomial coefficients, that is, the patterns and relations in Pascal’s triangle. At the suggestion of Jim Henle at Smith College, we considered a generalization of Pascal's triangle that is generated by the same rule as Pascal’s, except that the zeroth row is removed, and the pair of ones in the first row is replaced with any two finite numbers. Unlike a Pascal triangle, Henle triangles exhibit some asymmetry, and so many of the familiar properties of Pascal’s triangle do not hold.
In our research we made many comparisons between Pascal's triangle and the Henle triangles which led to some surprising similarities, and equally surprising differences. For example, the well known Erdos-Singmaster Conjecture, which in broad terms states that any number appears only finitely many times in Pascal's triangle, does not hold in a certain class of Henle triangles. Further we discovered that the Catalan numbers, a sequence of numbers that have a many rich applications in combinatorics, can be written as a combination of entries in Pascal's triangle and entries in a specific Henle triangle. We also found that Henle triangles share many of the more subtle and fascinating properties of Pascal's triangle.