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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.
2
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
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 20 2013
STATUS
approved