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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.

%I A261959 #22 Dec 17 2016 10:50:02
%S A261959 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 A261959 1,7,9,31,793,47293,1,1,1,7,9,31,403,4929,545835,1,1,1,7,9,31,403,
%U A261959 1597,33029,7087261,1,1,1,7,9,31,403,757,7913,388537,102247563
%N 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.
%H A261959 Alois P. Heinz, <a href="/A261959/b261959.txt">Antidiagonals n = 0..50, flattened</a>
%e A261959 A(3,1) = 7: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
%e A261959 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 A261959 Square array A(n,k) begins:
%e A261959 :    1,   1,   1,   1,   1,   1,   1, ...
%e A261959 :    1,   1,   1,   1,   1,   1,   1, ...
%e A261959 :    3,   1,   1,   1,   1,   1,   1, ...
%e A261959 :   13,   7,   7,   7,   7,   7,   7, ...
%e A261959 :   75,  21,   9,   9,   9,   9,   9, ...
%e A261959 :  541,  81,  31,  31,  31,  31,  31, ...
%e A261959 : 4683, 793, 403, 403, 403, 403, 403, ...
%p A261959 b:= proc(n, l) option remember; `if`(n=0, 1,
%p A261959        add(`if`(j in l, 0, binomial(n, j)*b(n-j,
%p A261959       `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
%p A261959     end:
%p A261959 A:= (n, k)-> b(n, [0$min(n,k)]):
%p A261959 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A261959 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 A261959 Columns k=0..6 give A000670, A114902, A261961, A272431, A272432, A272433, A272434.
%Y A261959 Main diagonal gives A032011.
%Y A261959 Cf. A261960.
%K A261959 nonn,tabl
%O A261959 0,6
%A A261959 _Alois P. Heinz_, Sep 06 2015