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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. 10
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A228637 A152795 A121585 * A257565 A276121 A262809

Adjacent sequences:  A261956 A261957 A261958 * A261960 A261961 A261962

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 06 2015

STATUS

approved

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Last modified March 18 16:20 EDT 2019. Contains 321292 sequences. (Running on oeis4.)