The principal numbers of K. Saito for the types image, image and image

The principal number image for a tree image is introduced by K. Saito [Principal image-cone for a tree image, Adv. Math. (2007), to appear, (Available online 16 February 2007)] as the maximal number of chambers contained in components of the complement of the graphic arrangement attached to image (see Section 1).∑l=1∞σ(Al)l!xl=tanx2+π4-1,∑l=3∞σ(Dl)l!xl=2(x-1)tanx2+π4-x2+2,∑l=4∞σ(El)l!xl=12x2-2x+3tanx2+π4-3x3-x-3.