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A250261
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Number A(n,k) of permutations p of [n] such that p(i) > p(i+1) iff i = 1 + k*m for some m >= 0; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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10
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1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 5, 1, 5, 1, 1, 1, 2, 3, 16, 1, 6, 1, 1, 1, 2, 3, 11, 61, 1, 7, 1, 1, 1, 2, 3, 4, 40, 272, 1, 8, 1, 1, 1, 2, 3, 4, 19, 99, 1385, 1, 9, 1, 1, 1, 2, 3, 4, 5, 78, 589, 7936, 1, 10, 1, 1, 1, 2, 3, 4, 5, 29, 217, 3194, 50521, 1, 11
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OFFSET
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0,10
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COMMENTS
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A(n,0) = A(n,k) for k>=n-1 and n>0.
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LINKS
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A. Mendes and J. Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 1, 2, 2, 2, 2, 2, 2, 2, ...
3, 1, 5, 3, 3, 3, 3, 3, 3, ...
4, 1, 16, 11, 4, 4, 4, 4, 4, ...
5, 1, 61, 40, 19, 5, 5, 5, 5, ...
6, 1, 272, 99, 78, 29, 6, 6, 6, ...
7, 1, 1385, 589, 217, 133, 41, 7, 7, ...
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MAPLE
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b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
`if`(t=1, add(b(u-j, o+j-1, irem(t+1, k), k), j=1..u),
add(b(u+j-1, o-j, irem(t+1, k), k), j=1..o)))
end:
A:= (n, k)-> b(0, n, 0, `if`(k=0, n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);
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MATHEMATICA
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b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, If[t == 1, Sum[ b[u-j, o+j-1, Mod[t+1, k], k], {j, 1, u}], Sum[ b[u+j-1, o-j, Mod[t+1, k], k], {j, 1, o}] ] ] ; A[n_, k_] := b[0, n, 0, If[k == 0, n, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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