The notion of a 12-representable graph was introduced by Jones et al. in [2]. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12representable graph is a comparability graph, and also that a tree is 12representable if and only if it is a double caterpillar. Moreover, Jones et al. initiated the study of 12-representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question in [2] is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs.In this paper we answer the question of Jones et al. by providing a complete characterization of 12-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area.

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Permutation tableaux were introduced by Steingrímsson and Williams. Corteel and Kim defined the sign of a permutation tableau in terms of the number of unrestricted columns. The sign-imbalance of permutation tableaux of length n is the sum of signs over permutation tableaux of length n. They have obtained a formula for the sign-imbalance of permutation tableaux of length n by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type B permutation tableaux introduced by Lam and Williams. We define a statistic wm over permutations and show that the number of unrestricted columns over permutation tableaux of length n is equally distributed with wm over permutations of length n. This leads to a combinatorial interpretation of the formula of Corteel and Kim. For type B permutation tableaux, we define the sign of a type B permutation tableau in term of the number of certain rows and columns. On the other hand, we construct a bijection between the type B permutation tableaux of length n and symmetric permutations of length 2n and we show that the statistic wm over symmetric permutations of length 2n is equally distributed with the number of certain rows and columns over type B permutation tableaux of length n. Based on this correspondence and an involution on symmetric permutation of length 2n, we obtain a sign-imbalance formula for type B permutation tableaux.

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