Plane Projections of the Dyck Triangle

4D Dyck triangle and its projections

The classic dyck triangle, the catalan triangle, and the catalan convolutionmatrix are plane projections of the multidimensional multidimensional dyck triangle.Each node is uniquely determined by two of four interrelated parameters: (i) the position of the current parenthesis, (ii) the current unbalance of the parentheses, (iii) the number of viewed left parentheses, and (iv) the same for right parentheses.The last two parameters can be redefined, respectively, as the index of the current catalan number and the index of the summand in the decomposition of the catalan number into the sum of squares (dyck squares).We consider six 2-dimensional projections (some of them are not yet in demand) and four 3-dimensional projections (some of them are not yet in demand) of the classic 4-dimensional (4d) dyck triangle.We also consider six 2-dimensional projections (some of them are not yet in demand) and four 3-dimensional projections (some of them are not yet in demand) of the classic 4-dimensional (4d) dyck triangle.