W. Lang, Feb 20 2009 A145575 tabf array: characteristic partition array for partitions with distinct parts. These partitions are called by W.L. fermionic. The array is filled (after the | mark) with 0s for k>=p(n)= A000041(n)). Partitions of n listed in Abramowitz-Stegun order p. 831-2 (see the main page for an A-number with the reference). n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... 1 1 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 1 1 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 1 1 0 0 1 0 0 0 0 0 | 0 0 0 0 0 0 0 0 0 0 0 7 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 8 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 | . . . n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 .. The next two rows, for n=9 and n=10, are (for k=1..p(n)=A000041(n)): n=9: [1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] . n=10: [1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. The row sums give A000009(n), n>=1: [1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ...] (total number of partititons of n with distnct parts). The array without zeros is A008289: a(n,m) tabf head (staircase) for A008289 (author: N. J. A. Sloane) n\m 1 2 3 4 5 .... 1 1 2 1 3 1 1 4 1 1 5 1 2 6 1 2 1 7 1 3 1 8 1 3 2 9 1 4 3 10 1 4 4 1 11 1 5 5 1 12 1 5 7 2 13 1 6 8 3 14 1 6 10 5 15 1 7 12 6 1 16 1 7 14 9 1 17 1 8 16 11 2 18 1 8 19 15 3 19 1 9 21 18 5 20 1 9 24 23 7 . . . The column nr. m starts with T(m)=m(m+1)/2 (triangular numbers A000217). The column sequences seem to coincide with: A000012, A004526, A001399, A001400, A001401, A001402, A026813 for m=1..7. (Compare with the array A145574(n,m) which has these columns, but with different n(m) start. ############################################### e.o.f. ##############################################################################