Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions

We prove a determinantal identity concerning Schur functions for 2-staircase diagrams lambda=(ln+l',ln,l(n-1)+l',l(n-1),...,l+l',l,l',0). When l=1 and l'=0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined enumerations of Alternating Sign Matrices.

[1]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[2]  Vladimir E. Korepin,et al.  Calculation of norms of Bethe wave functions , 1982 .

[3]  David P. Robbins,et al.  Alternating Sign Matrices and Descending Plane Partitions , 1983, J. Comb. Theory, Ser. A.

[4]  A. G. Izergin,et al.  Partition function of the six-vertex model in a finite volume , 1987 .

[5]  Greg Kuperberg,et al.  Another proof of the alternating sign matrix conjecture , 1996 .

[6]  Doron Zeilberger,et al.  Proof of the alternating sign matrix conjecture , 1994, Electron. J. Comb..

[7]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[8]  D. Bressoud.  Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture , 1999 .

[9]  Symmetric polynomials vanishing on thediagonals shifted by roots of unity , 2002, math/0209126.

[10]  A new way to deal with Izergin-Korepin determinant at root of unity , 2002, math-ph/0204042.

[11]  T. Miwa,et al.  Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials , 2002 .

[12]  Soichi Okada.  Enumeration of symmetry classes of alternating sign matrices and characters of classical groups , 2004 .

[13]  F. Colomo,et al.  On two-point boundary correlations in the six-vertex model with domain wall boundary conditions , 2005, math-ph/0503049.

[14]  A. Lascoux.  Pfaffians and representations of the symmetric group , 2006, math/0610510.

[15]  Yu. G. Stroganov,et al.  Izergin-Korepin Determinant at a Third Root of Unity , 2006 .

[16]  P. Zinn-Justin.  Combinatorial Point for Fused Loop Models , 2006, math-ph/0603018.

[17]  Paul Zinn-Justin,et al.  On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions , 2008, Electron. J. Comb..

[18]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[19]  R. Stanley.  Enumerative Combinatorics: Volume 1 , 2011 .