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A271466 Number T(n,k) of set partitions of [n] such that k is the largest element of the last block; triangle T(n,k), n>=1, 1<=k<=n, read by rows. 23
1, 0, 2, 0, 1, 4, 0, 1, 4, 10, 0, 1, 6, 15, 30, 0, 1, 10, 29, 59, 104, 0, 1, 18, 63, 139, 250, 406, 0, 1, 34, 149, 365, 692, 1145, 1754, 0, 1, 66, 375, 1039, 2110, 3627, 5649, 8280, 0, 1, 130, 989, 3149, 6932, 12521, 20085, 29874, 42294, 0, 1, 258, 2703, 10039, 24190, 46299, 77133, 117488, 168509, 231950 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Each set partition is written as a sequence of blocks, ordered by the smallest elements in the blocks.
LINKS
FORMULA
T(n,n) = 2 * A000110(n-1) = 2 * Sum_{j=0..n-1} T(n-1,j) for n>1.
EXAMPLE
T(1,1) = 1: 1.
T(2,2) = 2: 12, 1|2.
T(3,2) = 1: 13|2.
T(3,3) = 4: 123, 12|3, 1|23, 1|2|3.
T(4,2) = 1: 134|2.
T(4,3) = 4: 124|3, 14|23, 14|2|3, 1|24|3.
T(4,4) = 10: 1234, 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
T(5,2) = 1: 1345|2.
T(5,3) = 6: 1245|3, 145|23, 145|2|3, 14|25|3, 15|24|3, 1|245|3.
T(5,4) = 15: 1235|4, 125|34, 125|3|4, 12|35|4, 135|24, 135|2|4, 13|25|4, 15|234, 15|23|4, 1|235|4, 15|2|34, 1|25|34, 15|2|3|4, 1|25|3|4, 1|2|35|4.
Triangle T(n,k) begins:
1;
0, 2;
0, 1, 4;
0, 1, 4, 10;
0, 1, 6, 15, 30;
0, 1, 10, 29, 59, 104;
0, 1, 18, 63, 139, 250, 406;
0, 1, 34, 149, 365, 692, 1145, 1754;
0, 1, 66, 375, 1039, 2110, 3627, 5649, 8280;
0, 1, 130, 989, 3149, 6932, 12521, 20085, 29874, 42294;
...
MAPLE
b:= proc(n, m, c) option remember; `if`(n=0, x^c, add(
b(n-1, max(m, j), `if`(j>=m, n, c)), j=1..m+1))
end:
T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n-1))(b(n, 0$2)):
seq(T(n), n=1..12);
MATHEMATICA
b[n_, m_, c_] := b[n, m, c] = If[n == 0, x^c, Sum[b[n-1, Max[m, j], If[j >= m, n, c]], {j, 1, m+1}]];
T[n_] := Function[p, Table[Coefficient[p, x, n-i], {i, 0, n-1}]][b[n, 0, 0]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 24 2016, translated from Maple *)
CROSSREFS
Columns k=1-10 give: A000007(n-1), A054977(n-2), A052548(n-3) for n>3, A271743, A271744, A271745, A271746, A271747, A271748, A271749.
Main diagonal gives A186021(n-1).
Lower diagonals d=1-10 give: A271752, A271753, A271754, A271755, A271756, A271757, A271758, A271759, A271760, A271761.
Row sums give A000110.
T(2n,n) gives A271467.
T(2n+1,n+1) gives A271607.
Cf. A095149 (k is maximum of the first block), A113547 (k is minimum of the last block).
Sequence in context: A259873 A121462 A349706 * A218581 A307177 A340264
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 08 2016
STATUS
approved

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Last modified May 26 05:37 EDT 2024. Contains 372807 sequences. (Running on oeis4.)