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A294220 Number A(n,k) of ascent sequences of length n where no letter multiplicity is larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A294220 #21 Oct 19 2018 11:06:09
%S A294220 1,1,0,1,1,0,1,1,1,0,1,1,2,1,0,1,1,2,4,1,0,1,1,2,5,10,1,0,1,1,2,5,14,
%T A294220 27,1,0,1,1,2,5,15,47,83,1,0,1,1,2,5,15,52,180,277,1,0,1,1,2,5,15,53,
%U A294220 210,773,1015,1,0,1,1,2,5,15,53,216,964,3701,4007,1,0
%N A294220 Number A(n,k) of ascent sequences of length n where no letter multiplicity is larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A294220 Alois P. Heinz, <a href="/A294220/b294220.txt">Antidiagonals n = 0..20, flattened</a>
%H A294220 P. Duncan and Einar Steingrimsson, <a href="https://arxiv.org/abs/1109.3641">Pattern avoidance in ascent sequences</a>, arXiv:1109.3641, 2011
%F A294220 A(n,k) = Sum_{j=0..k} A294219(n,j).
%F A294220 A(n,k) = A(n,n) = A022493(n) for k >= n.
%e A294220 A(4,2) = 10: 0123, 0011, 0012, 0101, 0102, 0110, 0112, 0120, 0121, 0122.
%e A294220 Square array A(n,k) begins:
%e A294220   1, 1,    1,    1,    1,    1,    1,    1,    1, ...
%e A294220   0, 1,    1,    1,    1,    1,    1,    1,    1, ...
%e A294220   0, 1,    2,    2,    2,    2,    2,    2,    2, ...
%e A294220   0, 1,    4,    5,    5,    5,    5,    5,    5, ...
%e A294220   0, 1,   10,   14,   15,   15,   15,   15,   15, ...
%e A294220   0, 1,   27,   47,   52,   53,   53,   53,   53, ...
%e A294220   0, 1,   83,  180,  210,  216,  217,  217,  217, ...
%e A294220   0, 1,  277,  773,  964, 1006, 1013, 1014, 1014, ...
%e A294220   0, 1, 1015, 3701, 4960, 5270, 5326, 5334, 5335, ...
%p A294220 b:= proc(n, i, t, p, k) option remember; `if`(n=0, 1,
%p A294220       add(`if`(coeff(p, x, j)=k, 0, b(n-1, j, t+
%p A294220           `if`(j>i, 1, 0), p+x^j, k)), j=1..t+1))
%p A294220     end:
%p A294220 A:= (n, k)-> b(n, 0$3, min(n, k)):
%p A294220 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A294220 b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n == 0, 1, Sum[ If[ Coefficient[p, x, j] == k, 0, b[n-1, j, t + If[j>i, 1, 0], p + x^j, k]], {j, 1, t+1}]];
%t A294220 A[n_, k_] := b[n, 0, 0, 0, Min[n, k]];
%t A294220 Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* _Jean-François Alcover_, Aug 05 2018, translated from Maple *)
%Y A294220 Columns k=0-3 give: A000007, A000012, A202058, A317784.
%Y A294220 Main diagonal gives A022493.
%Y A294220 Cf. A294219.
%K A294220 nonn,tabl
%O A294220 0,13
%A A294220 _Alois P. Heinz_, Oct 25 2017