The sum of the areas of (2n + 2)-length Dyck paths, or total area, is equal to the number of points with ordinate 1 in Grand-Dyck paths of length 2n + 2, n ¿ 0. A bijective proof of this correspondence is shown by passing through an auxiliary class of marked paths. The sequence of numbers 1; 6; 29; 130; 562;::: counts the total area of (2n+2)-length Dyck paths as well as the number of points having ordinate 0 and which satisfy an additional condition, on 2n-length paths made up of rise and fall steps. First, a bijection between these points and the triangles constituting the total area of (2n+ 2)-length Dyck paths is established, and then the correspondence between the above-mentioned points and the points with ordinate 1 on (2n + 2)-length Grand-Dyck paths is shown.
Two bijections for the area of Dyck paths / E. PERGOLA. - In: DISCRETE MATHEMATICS. - ISSN 0012-365X. - STAMPA. - 241:(2001), pp. 435-447. [10.1016/S0012-365X(01)00129-7]
Two bijections for the area of Dyck paths
PERGOLA, ELISA
2001
Abstract
The sum of the areas of (2n + 2)-length Dyck paths, or total area, is equal to the number of points with ordinate 1 in Grand-Dyck paths of length 2n + 2, n ¿ 0. A bijective proof of this correspondence is shown by passing through an auxiliary class of marked paths. The sequence of numbers 1; 6; 29; 130; 562;::: counts the total area of (2n+2)-length Dyck paths as well as the number of points having ordinate 0 and which satisfy an additional condition, on 2n-length paths made up of rise and fall steps. First, a bijection between these points and the triangles constituting the total area of (2n+ 2)-length Dyck paths is established, and then the correspondence between the above-mentioned points and the points with ordinate 1 on (2n + 2)-length Grand-Dyck paths is shown.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.