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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).

%I A238352 #34 Feb 11 2015 06:05:29
%S A238352 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,
%T A238352 32,11,4,12,46,15,4,13,60,21,7,17,85,28,1,4,22,109,28,16,0,1,28,140,
%U A238352 51,7,5,34,179,57,26,1,42,239,74,25,5,48,300,107,24
%N 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).
%C A238352 Row sums are A000041.
%C A238352 Column k=0 is A238394, row sums over columns k>=1 give A238395.
%C A238352 T(A000217(k),k) = 1,
%C A238352 T(A000217(k),k-1) = 0 for k in {1, 3, 4, 5, ... },
%C A238352 T(A000217(k)-1,k-1) = k-1 for k>1.
%H A238352 Joerg Arndt and Alois P. Heinz, <a href="/A238352/b238352.txt">Rows n = 0..500, flattened</a>
%e A238352 Triangle starts:
%e A238352 00:   1;
%e A238352 01:   0,   1;
%e A238352 02:   1,   1;
%e A238352 03:   1,   1,   1;
%e A238352 04:   1,   4;
%e A238352 05:   2,   3,   2;
%e A238352 06:   3,   7,   0,  1;
%e A238352 07:   3,   7,   5;
%e A238352 08:   4,  14,   4;
%e A238352 09:   5,  19,   3,  3;
%e A238352 10:   8,  24,   9,  0,  1;
%e A238352 11:   9,  32,  11,  4;
%e A238352 12:  12,  46,  15,  4;
%e A238352 13:  13,  60,  21,  7;
%e A238352 14:  17,  85,  28,  1,  4;
%e A238352 15:  22, 109,  28, 16,  0, 1;
%e A238352 16:  28, 140,  51,  7,  5;
%e A238352 17:  34, 179,  57, 26,  1;
%e A238352 18:  42, 239,  74, 25,  5;
%e A238352 19:  48, 300, 107, 24, 11;
%e A238352 20:  59, 397, 122, 43,  1, 5;
%e A238352 21:  71, 495, 167, 37, 21, 0, 1;
%e A238352 ...
%e A238352 The 11 partitions of 6 together with their number of fixed points are:
%e A238352 01:  [ 1 1 1 1 1 1 ]   1
%e A238352 02:  [ 1 1 1 1 2 ]   1
%e A238352 03:  [ 1 1 1 3 ]   1
%e A238352 04:  [ 1 1 2 2 ]   1
%e A238352 05:  [ 1 1 4 ]   1
%e A238352 06:  [ 1 2 3 ]   3
%e A238352 07:  [ 1 5 ]   1
%e A238352 08:  [ 2 2 2 ]   1
%e A238352 09:  [ 2 4 ]   0
%e A238352 10:  [ 3 3 ]   0
%e A238352 11:  [ 6 ]   0
%e A238352 There are 3 partitions with no fixed points, 7 with one, none with 2, and one with 3, giving row 6.
%p A238352 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A238352       expand(b(n, i-1) +`if`(i>n, 0, (p-> add((c->c*x^j*
%p A238352       `if`(j=i, z, 1))(coeff(p, x, j)), j=0..degree(p,x)))
%p A238352       (x*b(n-i, i))))))
%p A238352     end:
%p A238352 T:= n-> (p->seq((q->add(coeff(q, x, j), j=0..degree(q, x)))
%p A238352     (coeff(p, z, i)), i=0..degree(p, z)))(b(n$2)):
%p A238352 seq(T(n), n=0..25);
%t A238352 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 *)
%Y A238352 Cf. A238349 (and A238350) for the same statistics for compositions.
%K A238352 nonn,tabf,look
%O A238352 0,10
%A A238352 _Joerg Arndt_ and _Alois P. Heinz_, Feb 26 2014