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