By Rüdiger and Baensch, Hans A. Riehl
Der Foto-Index ist ein selbstständiges, in sich abgeschlossenes Nachschlagewerk. In erster Linie soll er dem Benutzer der Aquarienatlanten 1-5 als Fotoreferenz mit Kurzbeschreibungen der Fischarten dienen. Der Index enthält ca. 20.000 Gattungs-/Art- oder Art-/Gattungsnamen von Fischen sowie ca. 4.600 Farbfotots. Insgesamt wurden ca. 4.000 Fischarten mit ihren Synonymen und deutschen Namen erfaßt.
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15. A combinatorial class is a set C together with a size function, wC : C −→ Z+ , (C, wC ) such that wC−1 (n) is ﬁnite for any n ∈ Z+ . Suppose (C, wC ) is a combinatorial class and c ∈ C. We call wC (c) the size of c and write simply w(c). There are two special combinatorial classes: E and Z which contain only one element of sizes 0 and 1, respectively. The subset of C which contains all the elements of size n, wC−1 (n), is denoted by Cn , and let Cn = |Cn |. The generating function of a combinatorial class (C, wC ) is given by z wC (c) = C(z) = c∈C Cn z n , n≥0 where Cn ⊂ C.
N = ∅ of standard Young diagrams, such that for 1 ≤ i ≤ n, μi is obtained from μi−1 by either adding one square or removing one square. For instance, the sequence is an oscillating tableaux; see Fig. 3. 1 k-Noncrossing partial matchings 25 Fig. 3. Oscillating tableaux: two subsequent shapes, μi−1 and μi , diﬀer by exactly one square. being sequences ∅ = μ0 , μ1 , . . , μn = ∅ such that for 1 ≤ i ≤ n, μi is obtained from μi−1 by either adding/removing one square or doing nothing; see Fig.
Then j is the element at the end of the ﬁrst row in Ti−1 . Otherwise suppose is the row of the square being removed from Ti−1 . Remove the square and insert its element x into the ( − 1)th row at precisely the position, where the removed element y would push it down via the RSK algorithm. That is, y is maximal subject to y < x. Since each column is strictly increasing y always exists. , a unique tableau Ti−1 . By construction, inserting j with the RSK algorithm produces Ti−1 . In Fig. 1. We shall furthermore see that ∗-tableaux can be interpreted as lattice walks.