A238352 Irregular triangle T(n,k) read by rows: T(n,k) is the number of partitions p(1), p(2), ..., p(m) of n (as weakly ascending list of parts) with k parts p at position p (fixed points), n>=0, 0<=k<= (column index of last nonzero term in row n).

1, 0, 1, 1, 1, 1, 1, 1, 1, 4, 2, 3, 2, 3, 7, 0, 1, 3, 7, 5, 4, 14, 4, 5, 19, 3, 3, 8, 24, 9, 0, 1, 9, 32, 11, 4, 12, 46, 15, 4, 13, 60, 21, 7, 17, 85, 28, 1, 4, 22, 109, 28, 16, 0, 1, 28, 140, 51, 7, 5, 34, 179, 57, 26, 1, 42, 239, 74, 25, 5, 48, 300, 107, 24

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Keywords

• nonn
• tabf
• look

Comments

Row sums are A000041.

Column k=0 is A238394, row sums over columns k>=1 give A238395.

T(A000217(k),k) = 1,

T(A000217(k),k-1) = 0 for k in {1, 3, 4, 5, ... },

T(A000217(k)-1,k-1) = k-1 for k>1.

Examples

			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.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..500, flattened

Crossrefs

Cf. A238349 (and A238350) for the same statistics for compositions.

Programs

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 *)