On the sign-imbalance of partition shapes

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.Comment: 12 pages. Better presentatio

New directions in enumerative chess problems

Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves in the solution is (usually) unique but the order is not, and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Almost all enumerative chess problems have been ``series-movers'', in which one side plays an uninterrupted series of moves, unanswered except possibly for one move by the opponent at the end. This can be convenient for setting up enumeration problems, but we show that other problem genres also lend themselves to composing enumerative problems. Some of the resulting enumerations cannot be shown (or have not yet been shown) in series-movers. This article is based on a presentation given at the banquet in honor of Richard Stanley's 60th birthday, and is dedicated to Stanley on this occasion.Comment: 14 pages, including many chess diagrams created with the Tutelaers font

On the sign-imbalance of skew partition shapes

Let the sign of a skew standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. We examine how the sign property is transferred by the skew Robinson-Schensted correspondence invented by Sagan and Stanley. The result is a remarkably simple generalization of the ordinary non-skew formula. The sum of the signs of all standard tableaux on a given skew shape is the sign-imbalance of that shape. We generalize previous results on the sign-imbalance of ordinary partition shapes to skew ones.Comment: 14 pages; former section 8 is removed and the rest is slightly update

Logical Notations on Chess

Namjena je rada pokušati prikazati šahovska pravila kao logičke iskaze te ljudsko (ne računalno) šahovsko razmišljanje kao postupak koji se može rekonstruirati pomoću klasičnih logičkih shema kao što su istinitosno stablo i deduktivni postupak u Fitchevu stilu. Šahovska pravila mogu se prevoditi na jezik logike prvoga reda, iako je mjestimično potrebno primjenjivati i operatore deontičke logike. Ciljevi se u šahovskoj igri mogu prikazati pomoću logičkog kvadrata. Mogućnosti jakih poteza općenito se mogu logički promišljati. Nesavršenost ljudske procjene kvalitete određenih poteza otvara mogućnost primjene elemenata trovrijednosnih logika. Ipak, rad je u većini posvećen mogućnostima primjene elementarne logike na osnove šahovske igre.This paper aims to present the rules of chess as logical statements and human (non-computer) chess reasoning as a decision procedure that can be reconstructed using classical logic methods such as tableaux and Fitch-style natural deductions. Chess rules can be represented in first-order logic terms with the occasional application of deontic logic operators. A consideration of chess game objectives can be offered by the tableaux method. In general terms, possible strong moves can be assessed with logical reasoning. Human imperfection in the quality assessment of certain moves opens up the possibility of applying three-valued logic parameters. However, the main focus of this paper is to examine the possibilities of applying elementary logic to the basics of chess play

Total positivity in loop groups I: whirls and curls

This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\R[t,t^{-1}]), and for the formal loop group GL_n(\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeroes of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the ``factorization problem'' using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlev\'{e} dynamical systems and Berenstein-Kazhdan's study of geometric crystals.Comment: 49 pages, 7 figure