To develop a bioartificial liver, long-term culture of fetal liver cells over a month's time was performed under three different culture conditions, i.e., stationary cultures and shaken-flask cultures, both by using a substratum made of porous polyvinyl formal (PVF) resin and conventional monolayer dish cultures as controls. Time course changes in cell numbers and albumin secretion were evaluated in cultures using Williams' E medium (WE) or minimum essential medium alpha (aMEM) supplemented with serum and hormones. In the WE medium, the numbers of fetal liver cells in all culture conditions gradually decreased with time, and albumin secretion rates rapidly decreased. In the stationary cultures using PVF, however, a significant increase in albumin secretion was observed after two weeks of culture. When cells were cultured in aMEM, the fetal liver cells exhibited sufficient proliferation in stationary and monolayer cultures, although albumin secretion rates per single cell were lower than those in WE. On the basis of these results, another series of culture experiments were performed, in which aMEM was used for the first 10 days to encourage cell proliferation, and the medium was changed to WE afterward. In these cultures, albumin secretion rates in the stationary cultures dramatically increased after the medium exchanges and were maintained at these high levels throughout the remaining culture period.

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It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length n is the Euler number En, alternating in sign, if n is odd (even, respectively). Josuat-Vergès obtained a q-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès and Kim.Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Vergès derived bivariate polynomials Qn(t, q) and Rn(t, q) as generalized Euler numbers via successive q-derivatives and multiplications by t on polynomials in t. The other goal in this paper is to give a combinatorial interpretation of Qn(t, q) and Rn(t, q) as the enumerators of the snakes with restrictions.

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.

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