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A276890 Number A(n,k) of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A276890 #16 Sep 21 2018 12:25:46
%S A276890 1,1,0,1,1,0,1,1,2,0,1,1,3,6,0,1,1,3,10,24,0,1,1,3,13,44,120,0,1,1,3,
%T A276890 13,62,234,720,0,1,1,3,13,75,352,1470,5040,0,1,1,3,13,75,466,2348,
%U A276890 10656,40320,0,1,1,3,13,75,541,3272,17880,87624,362880,0
%N A276890 Number A(n,k) of ordered set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A276890 Column k > 0 is asymptotic to exp(k-1) * n!. - _Vaclav Kotesovec_, Sep 22 2016
%H A276890 Alois P. Heinz, <a href="/A276890/b276890.txt">Antidiagonals n = 0..42, flattened</a>
%F A276890 A(n,k) = Sum_{j=0..k} A276891(n,j).
%e A276890 Square array A(n,k) begins:
%e A276890   1,    1,     1,     1,     1,     1,     1,     1, ...
%e A276890   0,    1,     1,     1,     1,     1,     1,     1, ...
%e A276890   0,    2,     3,     3,     3,     3,     3,     3, ...
%e A276890   0,    6,    10,    13,    13,    13,    13,    13, ...
%e A276890   0,   24,    44,    62,    75,    75,    75,    75, ...
%e A276890   0,  120,   234,   352,   466,   541,   541,   541, ...
%e A276890   0,  720,  1470,  2348,  3272,  4142,  4683,  4683, ...
%e A276890   0, 5040, 10656, 17880, 26032, 34792, 42610, 47293, ...
%p A276890 b:= proc(n, m, l) option remember; `if`(n=0, m!,
%p A276890       add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
%p A276890       `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
%p A276890     end:
%p A276890 A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0),
%p A276890              `if`(k=1, n!, b(n, 0, [0$(k-1)]))):
%p A276890 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A276890 b[n_, m_, l_List] := b[n, m, l] = If[n == 0, m!, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[k==0, If[n==0, 1, 0], If[k==1, n!, b[n, 0, Array[0&, k-1]]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jan 06 2017, after _Alois P. Heinz_ *)
%Y A276890 Columns k=0-10: A000007, A000142, A240172, A276893, A276894, A276895, A276896, A276897, A276898, A276899, A276900.
%Y A276890 Main diagonal gives: A000670.
%Y A276890 Cf. A276719, A276891.
%K A276890 nonn,tabl
%O A276890 0,9
%A A276890 _Alois P. Heinz_, Sep 21 2016