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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)
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)
We compute the generating function of column-strict plane partitions with parts in p rows of odd length and k parts equal to n. This refines both, Krattenthaler's [10] and the author's [5] refinement of the Bender-Knuth (ex-)Conjecture. The result is proved by an extension of the method for proving polynomial enumeration formulas which was introduced by the(More)