a(n,m) tabl head (triangle) for A139356 Digitsums of n in the factorial representation of n . a(n,m) tabf head (staircase) for A139365 n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ... 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 1 1 2 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 2 2 3 1 2 2 3 3 4 2 3 3 4 4 5 3 4 4 5 5 6 . . . n=5: [0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10] n=6: [0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 10, 11, 11, 12, 12, 13, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 10, 11, 11, 12, 12, 13, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 10, 11, 11, 12, 12, 13, 11, 12, 12, 13, 13, 14, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 7, 8, 8, 9, 9, 10, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 10, 11, 11, 12, 12, 13, 8, 9, 9, 10, 10, 11, 9, 10, 10, 11, 11, 12, 10, 11, 11, 12, 12, 13, 11, 12, 12, 13, 13, 14, 9, 10, 10, 11, 11, 12, 10, 11, 11, 12, 12, 13, 11, 12, 12, 13, 13, 14, 12, 13, 13, 14, 14, 15] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The (unique) factorial representation of numbers m from {0,1,...,n!-1}, n>=1, is m=sum(m(j)*j!,j=0..n-1) with m(j) from {0,1,...,n-j} (hence m(0)=0 always). For n=1..4 these representations, when written as [m(n-1),m(n-2),...,m(0)], are: n=1: [0] n=2: [0, 0], [1, 0] n=3: [0, 0, 0], [0, 1, 0], [1, 0, 0], [1, 1, 0], [2, 0, 0], [2, 1, 0] n=4: [0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 1, 1, 0], [0, 2, 0, 0], [0, 2, 1, 0], [1, 0, 0, 0], [1, 0, 1, 0], [1, 1, 0, 0], [1, 1, 1, 0], [1, 2, 0, 0], [1, 2, 1, 0], [2, 0, 0, 0], [2, 0, 1, 0], [2, 1, 0, 0], [2, 1, 1, 0], [2, 2, 0, 0], [2, 2, 1, 0], [3, 0, 0, 0], [3, 0, 1, 0], [3, 1, 0, 0], [3, 1, 1, 0], [3, 2, 0, 0], [3, 2, 1, 0] These are (D. N.) Lehmer codes for permutations of n objects {1,2,...,n} written in lexicographical order. ######################################### e.o.f. #########################################################