A232187 Number T(n,k) of parity alternating permutations of [n] with exactly k descents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

1, 1, 2, 1, 1, 5, 3, 2, 8, 2, 20, 44, 8, 6, 66, 66, 6, 114, 594, 414, 30, 24, 624, 1584, 624, 24, 864, 8784, 14544, 4464, 144, 120, 6840, 36240, 36240, 6840, 120, 8280, 147720, 471120, 353520, 55320, 840, 720, 86400, 857520, 1739520, 857520, 86400, 720, 96480

Offset

0,3

Comments

T(2n+1,k) = T(2n+1,n-k).

T(2n+2,n) = T(2n+1,n) + T(2n+3,n+1).

Links

Alois P. Heinz, Rows n = 0..29, flattened

Formula

T(2n+1,k) = n! * A173018(n+1,k) = A000142(n) * A173018(n+1,k).

Example

T(5,0) = 2: 12345, 34125.
T(5,1) = 8: 12543, 14325, 14523, 32145, 34521, 52143, 52341, 54123.
T(5,2) = 2: 32541, 54321.
T(6,2) = 8: 163254, 165432, 321654, 325416, 541632, 543216, 632541, 654321.
T(7,0) = 6: 1234567, 1256347, 3412567, 3456127, 5612347, 5634127.
T(7,1) = 66: 1234765, 1236547, 1236745, ..., 7456123, 7612345, 7634125.
T(7,2) = 66: 1254763, 1276543, 1432765, ..., 7652143, 7652341, 7654123.
T(7,3) = 6: 3254761, 3276541, 5432761, 5476321, 7632541, 7654321.
Triangle T(n,k) begins:
:  0 :   1;
:  1 :   1;
:  2 :   2;
:  3 :   1,    1;
:  4 :   5,    3;
:  5 :   2,    8,     2;
:  6 :  20,   44,     8;
:  7 :   6,   66,    66,     6;
:  8 : 114,  594,   414,    30;
:  9 :  24,  624,  1584,   624,   24;
: 10 : 864, 8784, 14544,  4464,  144;
: 11 : 120, 6840, 36240, 36240, 6840, 120;

Crossrefs

Column k=0 gives: A199660.

Row sums give: A092186 (for n>0).

T(2n+1,n) = A000142(n).

T(2n+2,n) = A001048(n+1).

Sequence in context: A259862 A182930 A372725 * A076241 A316399 A139347

Adjacent sequences: A232184 A232185 A232186 * A232188 A232189 A232190

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 20 2013

Status

approved