

A134434


Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k even entries that are followed by a smaller entry (n>=1, k>=0).


9



1, 1, 1, 4, 2, 4, 16, 4, 36, 72, 12, 36, 324, 324, 36, 576, 2592, 1728, 144, 576, 9216, 20736, 9216, 576, 14400, 115200, 172800, 57600, 2880, 14400, 360000, 1440000, 1440000, 360000, 14400, 518400, 6480000, 17280000, 12960000, 2592000, 86400
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OFFSET

1,4


COMMENTS

Row n has 1+floor(n/2) entries. T(2n1,0) = T(2n,0) = T(2n,n) = (n!)^2 = A001044(n).
This descent statistic is equidistributed on the symmetric group S_n with a multiplicative 2excedance statistic  see A136715 for details.  Peter Bala, Jan 23 2008


REFERENCES

S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173193.


LINKS

Table of n, a(n) for n=1..41.


FORMULA

T(2n,k) = [n!*C(n,k)]^2; T(2n+1,k) = [(n+1)!*C(n,k)]^2/(k+1). See the Kitaev & Remmel reference for recurrence relations (Sec. 3).


EXAMPLE

T(4,2) = 4 because we have 2143, 4213, 3421 and 4321.
Triangle starts:
1;
1, 1;
4, 2;
4, 16, 4;
36, 72, 12;
36, 324, 324, 36;


MAPLE

R[1]:=1: R[2]:=1+t: for n to 5 do R[2*n+1]:=sort(expand((1t)* (diff(R[2*n], t))+(2*n+1)*R[2*n])): R[2*n+2]:=sort(expand(t*(1t)*(diff(R[2*n+1], t))+(1+(2*n+1)*t)*R[2*n+1])) end do: for n to 11 do seq(coeff(R[n], t, j), j=0..floor((1/2)*n)); end do; # yields sequence in triangular form


CROSSREFS

Cf. A001044, A134435.
Cf. A136715.
Sequence in context: A011302 A302603 A085689 * A261254 A168613 A248251
Adjacent sequences: A134431 A134432 A134433 * A134435 A134436 A134437


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Nov 22 2007


STATUS

approved



