OFFSET
0,10
COMMENTS
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..500, flattened
EXAMPLE
Triangle starts:
00: 1;
01: 0, 1;
02: 1, 1;
03: 1, 1, 1;
04: 1, 4;
05: 2, 3, 2;
06: 3, 7, 0, 1;
07: 3, 7, 5;
08: 4, 14, 4;
09: 5, 19, 3, 3;
10: 8, 24, 9, 0, 1;
11: 9, 32, 11, 4;
12: 12, 46, 15, 4;
13: 13, 60, 21, 7;
14: 17, 85, 28, 1, 4;
15: 22, 109, 28, 16, 0, 1;
16: 28, 140, 51, 7, 5;
17: 34, 179, 57, 26, 1;
18: 42, 239, 74, 25, 5;
19: 48, 300, 107, 24, 11;
20: 59, 397, 122, 43, 1, 5;
21: 71, 495, 167, 37, 21, 0, 1;
...
The 11 partitions of 6 together with their number of fixed points are:
01: [ 1 1 1 1 1 1 ] 1
02: [ 1 1 1 1 2 ] 1
03: [ 1 1 1 3 ] 1
04: [ 1 1 2 2 ] 1
05: [ 1 1 4 ] 1
06: [ 1 2 3 ] 3
07: [ 1 5 ] 1
08: [ 2 2 2 ] 1
09: [ 2 4 ] 0
10: [ 3 3 ] 0
11: [ 6 ] 0
There are 3 partitions with no fixed points, 7 with one, none with 2, and one with 3, giving row 6.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
expand(b(n, i-1) +`if`(i>n, 0, (p-> add((c->c*x^j*
`if`(j=i, z, 1))(coeff(p, x, j)), j=0..degree(p, x)))
(x*b(n-i, i))))))
end:
T:= n-> (p->seq((q->add(coeff(q, x, j), j=0..degree(q, x)))
(coeff(p, z, i)), i=0..degree(p, z)))(b(n$2)):
seq(T(n), n=0..25);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[b[n, i-1] + If[i>n, 0, Function[{p}, Sum[Function[{c}, c*x^j* If[j == i, z, 1]][Coefficient[p, x, j]], {j, 0, Exponent[p, x]}]] [x*b[n-i, i]]]]]]; T[n_] := Function[{p}, Table[ Function[{q}, Sum[Coefficient[q, x, j], {j, 0, Exponent[q, x]}]][Coefficient[p, z, i]], {i, 0, Exponent[p, z]}]][b[n, n]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
CROSSREFS
KEYWORD
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 26 2014
STATUS
approved