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Doron Zeilberger

  1. Alexander Burstein's Lovely Combinatorial Proof of John Noonan's Beautiful Formula that the number of n-permutations that contain the Pattern 321 Exactly Once Equals (3/n)(2n)!/((n-3)!(n+3)!.

    Fri, 10/21/2011 - 15:01 — SomNambul
    Authors: Doron Zeilberger
    Subjects: Combinatorics
    Abstract

    In 1996, my brilliant student John Noonan, discovered, and proved that there
    are 3(2n)!/(n(n+3)!(n-3)!) ways to line-up n people of different heights in
    such a way that out of the n(n-1)(n-2)/6 possible triples of people exactly one
    is such that the tallest stands (not necessarily immediately) in front of the
    second-tallest, who in turn, stands (not necessarily immediately) in front of
    the shortest. In that article, I promised a prize of 25 dollars for a nice
    combinatorial proof. Alex Burstein gave such a proof. On Oct. 14, 2011, I
    talked about Alex's lovely proof at the Howard U.

  2. Balls in Boxes: Variations on a Theme of Warren Ewens and Herbert Wilf.

    Wed, 06/29/2011 - 15:01 — SomNambul
    Authors: Doron Zeilberger, Shalosh B. Ekhad
    Subjects: Statistics
    Abstract

    We comment on, elaborate, and extend the work of Warren Ewens and Herbert
    Wilf, described in their this http URL
    about the maximum in balls-and-boxes problem. In particular we meta-apply their
    ingenious method to show that it is not really needed, and that one is better
    off using the so-called Poisson Approximation, at least in applications to the
    real world, because extremely unlikely events mever happen in real life.

  3. The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that computed Ground States of Two-Electron Atoms (and its 2010 Redux).

    Wed, 06/02/2010 - 15:01 — SomNambul
    Authors: Doron Zeilberger, Christoph Koutschan
    Subjects: Symbolic Computation
    Abstract

    In order to appreciate how good we as mathematicians and scientists have it
    today, with extremely fast hardware and lots and lots of memory, as well as
    with readily available high-level software, both for numeric and symbolic
    computation, it may be a good idea to go back to the early days of electronic
    computers and carefully examine, as a case study, a problem that was considered
    a huge challenge back then, and compare notes. We chose C.L. Pekeris' 1958
    seminal work on the ground state energies of two-electron atoms.

  4. ${5\choose 2}$ Proofs that ${n\choose k} \leq {n\choose {k+1}}$ if $k<n/2$.

    Mon, 03/08/2010 - 16:01 — SomNambul
    Authors: Doron Zeilberger
    Subjects: Combinatorics
    Abstract

    There is no trivial mathematics, there are only trivial mathematicians! A
    mathematician is trivial if he or she believes that there exists trivial
    mathematics. Being a non-trivial mathematician myself, I will describe ten
    different proofs of the seemingly trivial fact that the number of ways of
    choosing k people out of n people is less than or equal to the number of ways
    of choosing k+1 people out of n people, provided that k is less than half of n.

  5. A Proof of George Andrews' and David Robbins' $q$-TSPP Conjecture.

    Wed, 02/24/2010 - 16:01 — SomNambul
    Authors: Doron Zeilberger, Manuel Kauers, Christoph Koutschan
    Subjects: Combinatorics
    Abstract

    The conjecture that the orbit-counting generating function for totally
    symmetric plane partitions can be written as an explicit product-formula, has
    been stated independently by George Andrews and David Robbins around 1983. We
    present a proof of this long-standing conjecture.

  6. Bijections for an identity of Young Tableaux.

    Tue, 02/23/2010 - 16:01 — SomNambul
    Authors: Doron Zeilberger, Amitai Regev
    Subjects: Combinatorics
    Abstract

    We present an elegant bijection between standard Young tableaux with 2n cells
    and at most two rows, and pairs of standard Young tableaux of the same shape,
    with n+1 cells, where only the top row can have more than one cell.

  7. A Eulogy for Jack Good.

    Thu, 12/03/2009 - 16:01 — SomNambul
    Authors: Doron Zeilberger
    Subjects: History and Overview
    Abstract

    Irving John ("Jack") Good (9 December 1916 - 5 April 2009) was one of my
    greatest heroes and influencers. On Oct. 25, 2009, I gave a twenty-three minute
    talk with the present title, and this article is an extended transcript of that
    talk. As with all my papers, the "accompanying" Maple package is much more
    important (mathematically, of course, I also talk about the human side of Jack
    in this article, and this is even more important than any math). In particular,
    I taught Jack's brilliant ideas to my computer, and now it can do even better
    than Jack.

  8. Finite Analogs of Szemer\'edi's Theorem.

    Wed, 10/28/2009 - 16:01 — SomNambul
    Authors: Doron Zeilberger, Paul Raff
    Subjects: Combinatorics
    Abstract

    One of the "deepest" theorems in mathematics is Endre Szemer\'edi's theorem
    about the inevitability of arithmetical progressions. Here we try to nibble at
    it, by doing "finite" analogs. This is already interesting for its own sake,
    but we believe that it has the potential to lead to extremely interesting
    sharpening of the currently rather weak bounds. Let's hope!

    Along with introducing the theory behind the finite analogs of Szemer\'edi's
    Theorem, full Maple, Mathematica, and Java code is provided. See paper for
    further details.

  9. A Multi-Set Identity for Partitions.

    Wed, 09/23/2009 - 13:19 — SomNambul
    Authors: Doron Zeilberger, Amitai Regev
    Subjects: Combinatorics
    Abstract

    We prove that the multiset {(RightArmLength,LeftArmLength)} ranging over all
    cells of all Ferrers diagrams with n cells equals the multiset
    {(RightArmLength,LegLength)} ranging over all cells of all Ferrers diagrams
    with n cells, thereby refining a multi-set identity proved by C. Bessenrodt and
    by Bacher and L. Manivel.

    Added In revised version: Guo-Niu Han kindly pointed out to us that our main
    result is contained in reference [B.H] of the present article.

  10. A Multi-Set Identity for Partitions.

    Wed, 09/23/2009 - 13:19 — SomNambul
    Authors: Doron Zeilberger, Amitai Regev
    Subjects: Combinatorics
    Abstract

    We prove that the multiset {(RightArmLength,LeftArmLength)} ranging over all
    cells of all Ferrers diagrams with n cells equals the multiset
    {(RightArmLength,LegLength)} ranging over all cells of all Ferrers diagrams
    with n cells, thereby refining a multi-set identity proved by C. Bessenrodt and
    by Bacher and L. Manivel.

    Added In revised version: Guo-Niu Han kindly pointed out to us that our main
    result is contained in reference [B.H] of the present article.

  11. The Mahonian probability distribution on words is asymptotically normal.

    Mon, 08/17/2009 - 18:30 — SomNambul
    Authors: E. Rodney Canfield, Svante Janson, Doron Zeilberger
    Subjects: Combinatorics
    Abstract

    The Mahonian statistic is the number of inversions in a permutation of a
    multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function
    for this statistic is the $q$ analog of the multinomial coefficient
    $\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating function is
    the normalization of the latter. We give two proofs that the distribution is
    asymptotically normal. The first is {\it computer-assisted}, based on the
    method of moments.

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