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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). 46
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
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.
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
Cf. A238349 (and A238350) for the same statistics for compositions.
Sequence in context: A278970 A182199 A217435 * A291357 A079636 A308261
KEYWORD
nonn,tabf,look
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 26 2014
STATUS
approved

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)