Combinatorial reciprocity for Monotone Triangles
Combinatorial reciprocity for Monotone Triangles
复制标题
单调三角形的组合互易
DOI：
10.1016/j.jcta.2013.04.002
复制
发表时间：
2011
期刊：
影响因子：
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通讯作者：
L. Riegler
中科院分区：
文献类型：
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作者：
Ilse Fischer;L. Riegler
下行k 1< k 2<⋯< k n的单调三角形的个数由n个变量中的多项式α （n; k 1，…，k n）给出。在弱递减序列k 1或k 2或或或k n处对该多项式的评估结果可以解释为称为递减单调三角形的新组合对象的有符号枚举。两类对象之间存在着惊人的联系，特别是证明了α (n; 1,2，…，n)= α （2n; n, n, n−1,n−1，…，1,1）。与单调三角形与交替符号矩阵的对应关系完美类比，底行为（n, n, n−1,n−1，…，1,1）的递减单调三角形集合与一类类asm矩阵的集合是一一对应的，该类矩阵在证明所声明的恒等代数上也起着重要的作用。找到一个客观的证明仍然是一个悬而未决的问题。
Abstract The number of Monotone Triangles with bottom row k 1< k 2<⋯< k n is given by a polynomial α (n; k 1,…, k n) in n variables. The evaluation of this polynomial at weakly decreasing sequences k 1⩾ k 2⩾⋯⩾ k n turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects–in particular it is shown that α (n; 1, 2,…, n)= α (2 n; n, n, n− 1, n− 1,…, 1, 1). In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row (n, n, n− 1, n− 1,…, 1, 1) is in one-to-one correspondence with a certain set of ASM-like matrices, which also play an important role in proving the claimed identity algebraically. Finding a bijective proof remains an open problem.
标记序多面体、单调三角形互易和部分着色的算术
DOI：
10.1137/130944849
发表时间：
2014
期刊：
SIAM J. Discret. Math.
影响因子：
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作者：
Katharina Jochemko;Raman Sanyal
通讯作者：
Raman Sanyal