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A047888 Rectangular array of numbers a(n,k) = number of permutations of n things with longest increasing subsequence of length <= k (1 <= k <= oo), read by antidiagonals. 8
1, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 14, 6, 2, 1, 1, 42, 23, 6, 2, 1, 1, 132, 103, 24, 6, 2, 1, 1, 429, 513, 119, 24, 6, 2, 1, 1, 1430, 2761, 694, 120, 24, 6, 2, 1, 1, 4862, 15767, 4582, 719, 120, 24, 6, 2, 1, 1, 16796, 94359, 33324, 5003, 720, 120, 24, 6, 2, 1, 1, 58786, 586590 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Also a(n,k) is the dimension of the space of SL(k)-invariants in V^n tensor (V^*)^n, where V is the standard k-dimensional representation of SL(k) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
LINKS
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53, no. 2, (1990), 257-285.
EXAMPLE
Square array a(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, ...
1, 5, 6, 6, 6, 6, ...
1, 14, 23, 24, 24, 24, ...
1, 42, 103, 119, 120, 120, ...
1, 132, 513, 694, 719, 720, ...
MATHEMATICA
rows = 12; h[l_List] := Module[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]] ; g[n_, i_, l_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := Table[g[n-k, Min[n-k, k], {k}], {k, 1, rows}] // Accumulate; A047888 = Table[T[n], {n, 1, rows}]; Table[A047888[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Alois P. Heinz *)
PROG
(PARI)
b(n, k) = {
my(x = 'x + O('x^(2*n)));
sum(i = 0, n, x^(2*i+k)/(i!*(i+k)!));
};
u(n, k) = {
my(v = Vec(matdet(matrix(k, k, i, j, b(n, abs(i-j))))));
return(vector((#v-1)\2, i, v[2*i+1] * i!^2));
};
A(n, k) = {
my(m = [; ]);
for (i = 1, k, m = concat(m, u(n, i)~));
return(m);
};
A(6, 6) \\ Gheorghe Coserea, Feb 02 2016
CROSSREFS
Rows of the array are partial sums of A047874. Cf. A047887.
Subarray of A214015.
Sequence in context: A098885 A106270 A319171 * A330964 A128704 A075259
KEYWORD
nonn,easy,nice,tabl
AUTHOR
Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane
EXTENSIONS
More terms from Naohiro Nomoto, Mar 01 2002
STATUS
approved

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Last modified March 19 07:14 EDT 2024. Contains 370954 sequences. (Running on oeis4.)