The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A319375 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows. 13
 1, 3, 1, 10, 4, 1, 35, 17, 7, 1, 136, 76, 36, 11, 1, 577, 357, 186, 81, 16, 1, 2682, 1737, 1023, 512, 162, 22, 1, 13435, 8997, 5867, 3151, 1345, 295, 29, 1, 72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1, 414761, 289253, 211888, 133853, 72973, 29503, 6676, 796, 46, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Alois P. Heinz, Rows n = 1..141, flattened Wikipedia, Partition of a set EXAMPLE The 5 set partitions of {1,2,3} are:   1   |2  |3   12  |3   13  |2   23  |1   123 so there are 10 elements in the first (largest) blocks, 4 in the second blocks and only 1 in the third blocks. Triangle T(n,k) begins:       1;       3,     1;      10,     4,     1;      35,    17,     7,     1;     136,    76,    36,    11,     1;     577,   357,   186,    81,    16,    1;    2682,  1737,  1023,   512,   162,   22,   1;   13435,  8997,  5867,  3151,  1345,  295,  29,  1;   72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1; MAPLE b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*       x^i, i=1..nops(l)), add(binomial(n-1, j-1)*       b(n-j, sort([l[], j])), j=1..n))     end: T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])): seq(T(n), n=1..12); # second Maple program: b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(        combinat[multinomial](n, i\$j, n-i*j)/j!*       b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))     end: T:= (n, k)-> b(n\$2, k)[2]: seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 02 2020 MATHEMATICA b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]]; Array[T, 12] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *) CROSSREFS Columns k=1-10 give: A097148, A333059, A333060, A333061, A333062, A333063, A333064, A333065, A333066, A333067. Row sums give A070071. Cf. A319298, A322384. Sequence in context: A280787 A126954 A176992 * A107870 A078817 A316193 Adjacent sequences:  A319372 A319373 A319374 * A319376 A319377 A319378 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 07 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)