A261959 Number A(n,k) of ordered set partitions of {1,2,...,n} such that no part has the same size as any of its k immediate predecessors; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 1, 1, 1, 7, 75, 1, 1, 1, 7, 21, 541, 1, 1, 1, 7, 9, 81, 4683, 1, 1, 1, 7, 9, 31, 793, 47293, 1, 1, 1, 7, 9, 31, 403, 4929, 545835, 1, 1, 1, 7, 9, 31, 403, 1597, 33029, 7087261, 1, 1, 1, 7, 9, 31, 403, 757, 7913, 388537, 102247563

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..50, flattened

Example

A(3,1) = 7: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
A(4,1) = 21: 1234, 1|234, 234|1, 2|134, 134|2, 3|124, 124|3, 4|123, 123|4, 3|12|4, 4|12|3, 2|13|4, 4|13|2, 2|14|3, 3|14|2, 1|23|4, 4|23|1, 1|24|3, 3|24|1, 1|34|2, 2|34|1.
Square array A(n,k) begins:
:    1,   1,   1,   1,   1,   1,   1, ...
:    1,   1,   1,   1,   1,   1,   1, ...
:    3,   1,   1,   1,   1,   1,   1, ...
:   13,   7,   7,   7,   7,   7,   7, ...
:   75,  21,   9,   9,   9,   9,   9, ...
:  541,  81,  31,  31,  31,  31,  31, ...
: 4683, 793, 403, 403, 403, 403, 403, ...

Maple

b:= proc(n, l) option remember; `if`(n=0, 1,
       add(`if`(j in l, 0, binomial(n, j)*b(n-j,
      `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
    end:
A:= (n, k)-> b(n, [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[MemberQ[l, j], 0, Binomial[n, j]*b[n-j, If[l == {}, {}, Append[ReplacePart[l, 1 -> Nothing], j]]]], {j, 1, n}]]; A[n_, k_] := b[n, Array[0&, Min[n, k]]];  Table[A[n, d-n], {d, 0, 10} , {n, 0, d}] // Flatten (* Jean-François Alcover, Dec 17 2016, after Alois P. Heinz *)

Crossrefs

Columns k=0..6 give A000670, A114902, A261961, A272431, A272432, A272433, A272434.

Main diagonal gives A032011.

Cf. A261960.

Sequence in context: A152795 A338817 A121585 * A348988 A257565 A276121

Adjacent sequences: A261956 A261957 A261958 * A261960 A261961 A261962

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 06 2015

Status

approved