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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.

%I A271466 #30 Sep 08 2021 07:15:30
%S A271466 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,
%T A271466 406,0,1,34,149,365,692,1145,1754,0,1,66,375,1039,2110,3627,5649,8280,
%U A271466 0,1,130,989,3149,6932,12521,20085,29874,42294,0,1,258,2703,10039,24190,46299,77133,117488,168509,231950
%N 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.
%C A271466 Each set partition is written as a sequence of blocks, ordered by the smallest elements in the blocks.
%H A271466 Alois P. Heinz, <a href="/A271466/b271466.txt">Rows n = 1..141, flattened</a>
%H A271466 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A271466 T(n,n) = 2 * A000110(n-1) = 2 * Sum_{j=0..n-1} T(n-1,j) for n>1.
%e A271466 T(1,1) = 1: 1.
%e A271466 T(2,2) = 2: 12, 1|2.
%e A271466 T(3,2) = 1: 13|2.
%e A271466 T(3,3) = 4: 123, 12|3, 1|23, 1|2|3.
%e A271466 T(4,2) = 1: 134|2.
%e A271466 T(4,3) = 4: 124|3, 14|23, 14|2|3, 1|24|3.
%e A271466 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.
%e A271466 T(5,2) = 1: 1345|2.
%e A271466 T(5,3) = 6: 1245|3, 145|23, 145|2|3, 14|25|3, 15|24|3, 1|245|3.
%e A271466 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.
%e A271466 Triangle T(n,k) begins:
%e A271466   1;
%e A271466   0, 2;
%e A271466   0, 1,   4;
%e A271466   0, 1,   4,  10;
%e A271466   0, 1,   6,  15,   30;
%e A271466   0, 1,  10,  29,   59,  104;
%e A271466   0, 1,  18,  63,  139,  250,   406;
%e A271466   0, 1,  34, 149,  365,  692,  1145,  1754;
%e A271466   0, 1,  66, 375, 1039, 2110,  3627,  5649,  8280;
%e A271466   0, 1, 130, 989, 3149, 6932, 12521, 20085, 29874, 42294;
%e A271466   ...
%p A271466 b:= proc(n, m, c) option remember; `if`(n=0, x^c, add(
%p A271466       b(n-1, max(m, j), `if`(j>=m, n, c)), j=1..m+1))
%p A271466     end:
%p A271466 T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n-1))(b(n, 0$2)):
%p A271466 seq(T(n), n=1..12);
%t A271466 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 A271466 T[n_] := Function[p, Table[Coefficient[p, x, n-i], {i, 0, n-1}]][b[n, 0, 0]];
%t A271466 Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Apr 24 2016, translated from Maple *)
%Y A271466 Columns k=1-10 give: A000007(n-1), A054977(n-2), A052548(n-3) for n>3, A271743, A271744, A271745, A271746, A271747, A271748, A271749.
%Y A271466 Main diagonal gives A186021(n-1).
%Y A271466 Lower diagonals d=1-10 give: A271752, A271753, A271754, A271755, A271756, A271757, A271758, A271759, A271760, A271761.
%Y A271466 Row sums give A000110.
%Y A271466 T(2n,n) gives A271467.
%Y A271466 T(2n+1,n+1) gives A271607.
%Y A271466 Cf. A095149 (k is maximum of the first block), A113547 (k is minimum of the last block).
%K A271466 nonn,tabl
%O A271466 1,3
%A A271466 _Alois P. Heinz_, Apr 08 2016