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A269129
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Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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21
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0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 43, 23, 1, 0, 0, 1, 374, 1879, 119, 1, 0, 0, 1, 3199, 173891, 102011, 719, 1, 0, 0, 1, 26945, 16140983, 117392909, 7235651, 5039, 1, 0, 0, 1, 224296, 1474050783, 142951955371, 117108036719, 674641325, 40319, 1
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OFFSET
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0,14
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LINKS
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J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905
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FORMULA
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EXAMPLE
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Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
1, 5, 43, 374, 3199, 26945, ...
1, 23, 1879, 173891, 16140983, 1474050783, ...
1, 119, 102011, 117392909, 142951955371, 173996758190594, ...
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MAPLE
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g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
binomial(n-1, l[-1]-1)+add(f(sort(subsop(j=l[j]
-1, l))), j=1..nops(l)-1))(add(i, i=l))
end:
f:= l->(n->`if`(n=0, 1, `if`(l[1]=0, 0, `if`(n=1 or l[-1]=1, 1,
`if`(n=2, binomial(l[1]+l[2], l[1])-1, g(l))))))(nops(l)):
A:= (n, k)-> (k*n)!/k!^n - f([k$n]):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(k, p, j, l, t) option remember;
`if`(k=0, (-1)^t/l!, `if`(p<0, 0, add(b(k-i, p-1,
j+1, l+i*j, irem(t+i*j, 2))/(i!*p!^i), i=0..k)))
end:
A:= (n, k)-> (n*k)!*(1/k!^n-b(n, k-1, 1, 0, irem(n, 2))*n!):
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MATHEMATICA
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b[k_, p_, j_, l_, t_] := b[k, p, j, l, t] = If[k == 0, (-1)^t/l!, If[p < 0, 0, Sum[b[k-i, p-1, j+1, l + i j, Mod[t + i j, 2]]/(i! p!^i), {i, 0, k}]] ];
A[n_, k_] := (n k)! (1/k!^n - b[n, k-1, 1, 0, Mod[n, 2]] n!); Table[ Table[ A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A057427, A033312, A267532, A269113, A269114, A269115, A269116, A269117, A269118, A269119, A269120.
Rows n=0-10 give: A000004, A000007, A000012, A269121, A269122, A269123, A269124, A269125, A269126, A269127, A269128.
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KEYWORD
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AUTHOR
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STATUS
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approved
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