scholarly journals Combinatorial Proof of a Curious $q$-Binomial Coefficient Identity

10.37236/462 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Binomial Coefficient ◽  
Combinatorial Proof ◽  
Binomial Coefficient Identity

Using the Algorithm Z developed by Zeilberger, we give a combinatorial proof of the following $q$-binomial coefficient identity $$ \sum_{k=0}^m(-1)^{m-k}{m\brack k}{n+k\brack a}(-xq^a;q)_{n+k-a}q^{{k+1\choose 2}-mk+{a\choose 2}} $$ $$=\sum_{k=0}^n{n\brack k}{m+k\brack a}x^{m+k-a}q^{mn+{k\choose 2}}, $$ which was obtained by Hou and Zeng [European J. Combin. 28 (2007), 214–227].

scholarly journals A Binomial Coefficient Identity Associated to a Conjecture of Beukers

10.37236/1348 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Scott Ahlgren ◽  
Shalosh B. Ekhad ◽  
Ken Ono ◽  
Doron Zeilberger
Keyword(s):  
Binomial Coefficient ◽  
Wz Method ◽  
Binomial Coefficient Identity

Using the WZ method, a binomial coefficient identity is proved. This identity is noteworthy since its truth is known to imply a conjecture of Beukers.

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

A Modular Binomial Coefficient Identity: 11118

2006 ◽  
Vol 113 (7) ◽  
pp. 657
Author(s):  
Vesselin Dimitrov ◽  
Robin Chapman
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficient Identity

scholarly journals Counting $k$-Marked Durfee Symbols

10.37236/528 ◽  
2010 ◽  
Vol 18 (1) ◽  
Author(s):  
Kağan Kurşungöz
Keyword(s):  
Generating Functions ◽  
Binomial Coefficient ◽  
Equivalence Classes ◽  
Alternative Characterization ◽  
Binomial Coefficient Identity

An alternative characterization of $k$-marked Durfee symbols defined by Andrews is given. Some identities involving generating functions of $k$-marked Durfee symbols are proven combinatorially by considering the symbols not individually, but in equivalence classes. Also, a related binomial coefficient identity is obtained in the course.

scholarly journals Unimodality via Alternating Gamma Vectors

10.37236/5950 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Charles Brittenham ◽  
Andrew T. Carroll ◽  
T. Kyle Petersen ◽  
Connor Thomas
Keyword(s):  
Binomial Coefficient ◽  
The Other ◽  
Combinatorial Proof ◽  
Sufficient Condition ◽  
Proof Of Concept ◽  
Positive Formula

For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative $g$-vector. A sufficient condition for unimodality is having a nonnegative $\gamma$-vector, though one can have negative entries in the $\gamma$-vector and still have a nonnegative $g$-vector.In this paper we provide combinatorial models for three families of $\gamma$-vectors that alternate in sign. In each case, the $\gamma$-vectors come from unimodal polynomials with straightforward combinatorial descriptions, but for which there is no straightforward combinatorial proof of unimodality. By using the transformation from $\gamma$-vector to $g$-vector, we express the entries of the $g$-vector combinatorially, but as an alternating sum. In the case of the $q$-analogue of $n!$, we use a sign-reversing involution to interpret the alternating sum, resulting in a manifestly positive formula for the $g$-vector. In other words, we give a combinatorial proof of unimodality. We consider this a "proof of concept" result that we hope can inspire a similar result for the other two cases, $\prod_{j=1}^n (1+q^j)$ and the $q$-binomial coefficient ${n\brack k}$.

Another Binomial Coefficient Identity: 10878

2003 ◽  
Vol 110 (4) ◽  
pp. 342
Author(s):  
Pal Peter Dalyay ◽  
Said Amghibech ◽  
David Callan
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficient Identity

scholarly journals A combinatorial proof of Cayley's theorem on Pfaffians

1966 ◽  
Vol 1 (2) ◽  
pp. 224-232 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Combinatorial Proof

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

Sign in / Sign up

Export Citation Format

Share Document