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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

%I #22 Dec 17 2016 10:50:02

%S 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,

%T 1,7,9,31,793,47293,1,1,1,7,9,31,403,4929,545835,1,1,1,7,9,31,403,

%U 1597,33029,7087261,1,1,1,7,9,31,403,757,7913,388537,102247563

%N 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.

%H Alois P. Heinz, <a href="/A261959/b261959.txt">Antidiagonals n = 0..50, flattened</a>

%e A(3,1) = 7: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.

%e 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.

%e Square array A(n,k) begins:

%e : 1, 1, 1, 1, 1, 1, 1, ...

%e : 1, 1, 1, 1, 1, 1, 1, ...

%e : 3, 1, 1, 1, 1, 1, 1, ...

%e : 13, 7, 7, 7, 7, 7, 7, ...

%e : 75, 21, 9, 9, 9, 9, 9, ...

%e : 541, 81, 31, 31, 31, 31, 31, ...

%e : 4683, 793, 403, 403, 403, 403, 403, ...

%p b:= proc(n, l) option remember; `if`(n=0, 1,

%p add(`if`(j in l, 0, binomial(n, j)*b(n-j,

%p `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))

%p end:

%p A:= (n, k)-> b(n, [0$min(n,k)]):

%p seq(seq(A(n, d-n), n=0..d), d=0..10);

%t 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_ *)

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

%Y Main diagonal gives A032011.

%Y Cf. A261960.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Sep 06 2015