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Search: id:A134434
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| A134434 |
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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). |
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+0
10
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| 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; listen)
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OFFSET
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1,4
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COMMENT
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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 (pbala(AT)toucansurf.com), Jan 23 2008
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REFERENCES
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S. Kitaev and J. Remmel, Classifying descents according to parity, Annals of Combinatorics, 11, 2007, 173-193.
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FORMULA
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T(2n,k)=[n!*binom(n,k)]^2; T(2n+1,k)=[(n+1)!*binom(n,k)]^2/(k+1). See the Kitaev & Remmel reference for recurrence relations (Sec. 3).
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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Cf. A001044, A134435.
Cf. A136715.
Sequence in context: A011382 A011302 A085689 this_sequence A139809 A094099 A107046
Adjacent sequences: A134431 A134432 A134433 this_sequence A134435 A134436 A134437
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 22 2007
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