|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'506'133
Articles rated: 2609
27 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
On Two Families of Generalizations of Pascal's Triangle | | Michael A. Allen
; Kenneth Edwards
; | | Date: |
31 Jan 2022 | | Abstract: | We consider two families of Pascal-like triangles that have all ones on the
left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases
are Pascal’s triangle and the two families also coincide when $m=2$. Members of
the first family obey Pascal’s recurrence everywhere inside the triangle. We
show that the $m$-th triangle can also be obtained by reversing the elements up
to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$
Riordan array. Properties of this family of triangles can be obtained quickly
as a result. The $(n,k)$-th entry in the $m$-th member of the second family of
triangles is the number of tilings of an $(n+k) imes1$ board that use $k$
$(1,m-1)$-fences and $n-k$ unit squares. A $(1,g)$-fence is composed of two
unit square sub-tiles separated by a gap of width $g$. We show that the entries
in the antidiagonals of these triangles are coefficients of products of powers
of two consecutive Fibonacci polynomials and give a bijective proof that these
coefficients give the number of $k$-subsets of ${1,2,ldots,n-m}$ such that
no two elements of a subset differ by $m$. Other properties of the second
family of triangles are also obtained via a combinatorial approach. Finally, we
give necessary and sufficient conditions for any Pascal-like triangle (or its
row-reversed version) derived from tiling $(n imes1)$-boards to be a Riordan
array. | | Source: | arXiv, 2201.13253 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER