Analogous to MacMahon’s definition of compositions, the second author, 2000, defined   n     n       -color compositions as   n     n       -color ordered partitions. Several combinatorial properties of   n     n       -color compositions were subsequently found by him, 2003, G. Narang and A.K. Agarwal, 2006, 2008, and C. Shapcott, 2012. Recently, Y. Guo 2012, 2013, studied three restricted   n     n       -color composition functions and found generating functions, explicit formulas and recurrence relations for them. In this paper, we introduce nine new restricted   n     n       -color composition functions. It is then shown that three of them are equal to three other functions. Eventually, here we study six integer sequences from the combinatorial point of view. In addition to generating functions, explicit formulas and recurrence relations for each of them, we also obtain six combinatorial identities involving different   n     n       -color composition functions. The main results of this paper are contained in Section 3.
['R. Sachdeva', 'A.K. Agarwal']
Combinatorics of certain restricted n-color composition functions