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A213275
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Number A(n,k) of words w where each letter of the k-ary alphabet occurs n times and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(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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19
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 7, 1, 1, 1, 120, 105, 106, 19, 1, 1, 1, 720, 945, 2575, 1075, 56, 1, 1, 1, 5040, 10395, 87595, 115955, 13326, 174, 1, 1, 1, 40320, 135135, 3864040, 19558470, 7364321, 188196, 561, 1, 1
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OFFSET
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0,8
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COMMENTS
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The words counted by A(n,k) have length n*k.
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..20, flattened
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EXAMPLE
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A(0,k) = A(n,0) = 1: the empty word.
A(n,1) = 1: (a)^n for alphabet {a}.
A(1,2) = 2: ab, ba for alphabet {a,b}.
A(1,3) = 6: abc, acb, bac, bca, cab, cba for alphabet {a,b,c}.
A(2,2) = 3: aabb, abab, baab.
A(2,3) = 15: aabbcc, aabcbc, aacbbc, ababcc, abacbc, abcabc, acabbc, acbabc, baabcc, baacbc, bacabc, bcaabc, caabbc, cababc, cbaabc.
A(3,2) = 7: aaabbb, aababb, aabbab, abaabb, ababab, baaabb, baabab.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 6, 24, 120, 720, ...
1, 1, 3, 15, 105, 945, 10395, ...
1, 1, 7, 106, 2575, 87595, 3864040, ...
1, 1, 19, 1075, 115955, 19558470, 4622269345, ...
1, 1, 56, 13326, 7364321, 7236515981, 10915151070941, ...
1, 1, 174, 188196, 586368681, 3745777177366, 40684710729862072, ...
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MAPLE
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A:= (n, k)-> b([n$k]):
b:= proc(l) option remember;
`if`({l[]} minus {0}={}, 1, add(`if`(g(l, i),
b(subsop(i=l[i]-1, l)), 0), i=1..nops(l)))
end:
g:= proc(l, i) local j;
if l[i]<1 then return false
elif l[i]>1 then for j from i+1 to nops(l) do
if l[i]<=l[j] then return false
elif l[j]>0 then break
fi od fi; true
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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a[n_, k_] := b[Array[n&, k]];
b[l_] := b[l] = If[l ~Complement~ {0} == {}, 1, Sum[If[g[l, i], b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, Length[l]}]];
g[l_, i_] := Module[{j},
If[l[[i]] < 1, Return[False],
If[l[[i]] > 1, For[j = i+1, j <= Length[l], j++,
If[l[[i]] <= l[[j]], Return[False],
If[l[[j]] > 0, Break[]]]]]]; True];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
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CROSSREFS
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Rows n=0-10 give: A000012, A000142, A001147, A213863, A213864, A213865, A213866, A213867, A213868, A213869, A213870.
Columns k=0+1, 2-10 give: A000012, A005807(n-1) for n>0, A213873, A213874, A213875, A213876, A213877, A213878, A213871, A213872.
Main diagonal gives A213862.
Cf. A213276.
Sequence in context: A229557 A332700 A256268 * A069777 A225816 A227655
Adjacent sequences: A213272 A213273 A213274 * A213276 A213277 A213278
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Jun 08 2012
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STATUS
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approved
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