Abstract
Starting from the two-species partially asymmetric simple exclusion process, we study a subclass of signed permutations, the partially signed permutations, using the combinatorics of Laguerre histories. From this physical and bijective point of view, we obtain a natural descent statistic on partially signed permutations; as well as partially signed permutations patterns.
Highlights
The two-species partially asymmetric simple exclusion process (2-PASEP) is a Markov chain with two types of particles and holes (◦)
If there are no particles of type, we recover the classical PASEP
The classical PASEP has given rise to beautiful combinatorics related to Laguerre histories [16], permutations [5, 16, 25], permutation tableaux [8, 7, 25], alternative tableaux [20] and staircase tableaux [9] in its most general case
Summary
The two-species partially asymmetric simple exclusion process (2-PASEP) is a Markov chain with two types of particles ( and ) and holes (◦). In the case of the 2-PASEP the Matrix Ansatz extends naturally [27] We generalize the results of [5, 16] related to Laguerre histories and permutations In these cases the states of the PASEP are in bijection with compositions and the statistic coming from the PASEP is in bijection with the weight of the paths or equivalently the number of the generalized patterns (31 − 2) of the permutation. The states of the 2-PASEP are in bijection with segmented compositions and the statistic coming from the 2-PASEP is in bijection with the weight of the paths or equivalently the number of some generalized patterns of the partially signed permutation.
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