A229892 - OEIS

A229892

Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 5, 3, 1, 1, 0, 16, 6, 4, 1, 1, 0, 61, 26, 10, 5, 1, 1, 0, 272, 71, 20, 15, 6, 1, 1, 0, 1385, 413, 125, 35, 21, 7, 1, 1, 0, 7936, 1456, 461, 70, 56, 28, 8, 1, 1, 0, 50521, 10576, 1301, 574, 126, 84, 36, 9, 1, 1

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OFFSET

0,8

COMMENTS

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = T(n,n) = A000012(n) = 1 for k>n.

T(2*n,n) = C(2*n-1,n) = A088218(n) = A001700(n-1) for n>0.

T(2*n+1,n) = C(2*n,n) = A000984(n).

T(2*n+1,n+1) = C(2n,n-1) = A001791(n) for n>0.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

T(7,3) = 20: 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321.

EXAMPLE

Triangle T(n,k) begins:

1, 1;

0, 1, 1;

0, 2, 1, 1;

0, 5, 3, 1, 1;

0, 16, 6, 4, 1, 1;

0, 61, 26, 10, 5, 1, 1;

0, 272, 71, 20, 15, 6, 1, 1;

0, 1385, 413, 125, 35, 21, 7, 1, 1;

MAPLE

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,

b(o-j, u+j-1, 1, k), b(u+j-1, o-j, t+1, k)), j=1..o))

end:

T:= (n, k)-> `if`(k+1>=n, 1, `if`(k=0, 0, b(0, n, 0, k))):

seq(seq(T(n, k), k=0..n), n=0..10);

MATHEMATICA

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == k, b[o-j, u+j-1, 1, k], b[u+j-1, o-j, t+1, k]], {j, 1, o}]]; t[n_, k_] := If[k+1 >= n, 1, If[k == 0, 0, b[0, n, 0, k]]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

CROSSREFS

Columns k=1-10 give: A000111, A058258, A229884, A229885, A229886, A229887, A229888, A229889, A229890, A229891.

Cf. A227941, A229066, A229551.

Sequence in context: A198237 A122049 A238802 * A064879 A173591 A343320

Adjacent sequences: A229889 A229890 A229891 * A229893 A229894 A229895

KEYWORD

nonn,tabl,eigen

AUTHOR

Alois P. Heinz, Oct 02 2013

STATUS

approved