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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row n has 1+floor(n/2) entries. T(2n-1,0) = T(2n,0) = T(2n,n) = (n!)^2 = A001044(n).

This descent statistic is equidistributed on the symmetric group S_n with a multiplicative 2-excedance 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, 173-193.

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((1-t)* (diff(R[2*n], t))+(2*n+1)*R[2*n])): R[2*n+2]:=sort(expand(t*(1-t)*(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

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Last modified January 21 03:16 EST 2021. Contains 340332 sequences. (Running on oeis4.)