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In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of n × n alternating sign matrices with a 1 at the top of the i-th column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we… (More)

We show that a graph has an orientation under which every circuit of even length is clockwise odd if and only if the graph contains no subgraph which is, after the contraction of at most one circuit of odd length, an even subdivision of K 2,3. In fact we give a more general characterisation of graphs that have an orientation under which every even circuit… (More)

- Ilse Fischer, Dan Romik
- 2009

We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general " d-refined " enumerations of ASMs according to the first d rows. For the doubly-refined case of d = 2, we derive a system of linear equations satisfied by the doubly-refined… (More)

We compute the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the 'almost central' rhombus above the centre.

- ILSE FISCHER
- 2009

Monotone triangles are plane integer arrays of triangular shape with certain mono-tonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n × n alternating sign matrices when prescribing (1, 2,. .. , n) as bottom row of the array. We define monotone (d, m)–trapezoids as monotone triangles with m… (More)

1 The second author thanks the University of Klagenfurt for its hospitality while this research was undertaken. Abstract A graph is 1-extendible if every edge has a 1-factor containing it. A 1-extendible non-bipartite graph G is said to be near bipartite if there exist edges e 1 and e 2 such that G − {e 1 , e 2 } is 1-extendible and bipartite. We… (More)

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with… (More)