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

Author

Alois P. Heinz, Nov 20 2013

Keywords

• nonn
• tabf

Comments

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

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

Examples

			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;
		

Links

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

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).

Formula

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