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A240608
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Number A(n,k) of n-length words w over a k-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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11
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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, 7, 1, 0, 1, 1, 2, 5, 13, 14, 1, 0, 1, 1, 2, 5, 14, 35, 25, 1, 0, 1, 1, 2, 5, 14, 45, 94, 50, 1, 0, 1, 1, 2, 5, 14, 46, 149, 254, 91, 1, 0, 1, 1, 2, 5, 14, 46, 164, 509, 688, 182, 1, 0, 1, 1, 2, 5, 14, 46, 165, 629, 1756, 1872, 336, 1, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,13
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LINKS
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 4, 5, 5, 5, 5, 5, 5, ...
0, 1, 7, 13, 14, 14, 14, 14, 14, ...
0, 1, 14, 35, 45, 46, 46, 46, 46, ...
0, 1, 25, 94, 149, 164, 165, 165, 165, ...
0, 1, 50, 254, 509, 629, 650, 651, 651, ...
0, 1, 91, 688, 1756, 2511, 2742, 2770, 2771, ...
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MAPLE
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b:= proc(n, k, l) option remember; `if`(n=0, 1, `if`(nops(l)<k,
b(n-1, k, [l[], 1]), 0) +add(`if`(i=1 or l[i]<=l[i-1],
b(n-1, k, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))
end:
A:= (n, k)-> b(n, min(k, n), []):
seq(seq(A(n, d-n), n=0..d), d=0..14);
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MATHEMATICA
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b[n_, k_, l_List] := b[n, k, l] = If[n == 0, 1, If[Length[l]<k, b[n-1, k, Append[l, 1]], 0] + Sum[If[i == 1 || l[[i]] <= l[[i-1]], b[n-1, k, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}]]; A[n_, k_] := b[n, Min[k, n], {}]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000012, A026010(n-1) for n>0, A240609, A240610, A240611, A240612, A240613, A240614, A240615, A240616.
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KEYWORD
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AUTHOR
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STATUS
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approved
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