Refined enumerations of alternating sign matrices: monotone ( d, m)-trapezoids with prescribed top and bottom row
Ilse Fischer
DOI: 10.1007/s10801-010-0243-7
Abstract
Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n\times n alternating sign matrices when prescribing (1,2,\cdots , n) as bottom row of the array. We define monotone ( d, m)-trapezoids as monotone triangles with m rows where the d - 1 top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row ( k 1,\cdots , k n ) is given by a polynomial α ( n; k 1,\cdots , k n ) in the k i 's. The main purpose of this paper is to show that the number of monotone ( d, m)-trapezoids with prescribed top and bottom row appears as a coefficient in the expansion of a specialisation of α ( n; k 1,\cdots , k n ) with respect to a certain polynomial basis. This settles a generalisation of a recent conjecture of Romik et al. (Adv. Math. 222:2004-2035, 2009). Among other things, the result is used to express the number of monotone triangles with bottom row (1,2,\cdots , i - 1, i+1,\cdots , j - 1, j+1,\cdots , n) (which is, by the standard bijection, also the number of n\times n alternating sign matrices with given top two rows) in terms of the number of n\times n alternating sign matrices with prescribed top and bottom row, and, by a formula of Stroganov for the latter numbers, to provide an explicit formula for the first numbers. (A formula of this type was first derived by Karklinsky and Romik using the relation of alternating sign matrices to the six-vertex model.)
Pages: 239–257
Keywords: keywords exact enumeration; alternating sign matrices; monotone triangles
Full Text: PDF
References
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2. Colomo, F., Pranko, A.G.: On two-point boundary correlations in the six-vertex model with DWBC. J. Stat. Mech. 05010 (2005)
3. Fischer, I.: A new proof of the refined alternating sign matrix theorem. J. Comb. Theory: Ser. A 114, 253-264 (2007)
4. Fischer, I.: The operator formula for monotone triangles-simplified proof and three generalizations. J. Comb. Theory: Ser. A (in press).
5. Fischer, I., Romik, D.: More refined enumerations of alternating sign matrices. Adv. Math. 222, 2004- 2035 (2009)
6. Karklinsky, M., Romik, D.: A formula for the doubly refined enumeration of alternating sign matrices. Adv. Appl. Math. 45, 28-35 (2010)
7. Mills, W.H., Robbins, D.P., Rumsey, H.: Alternating sign matrices and descending plane partitions. J. Comb. Theory: Ser. A 34, 340-359 (1983)
8. Stroganov, Yu.G.: A new way to deal with Izergin-Korepin determinant at root of unity.
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