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A145882
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Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} having k descents. (n>=1, k>=0).
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1
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1, 1, 1, 2, 1, 5, 5, 1, 1, 14, 30, 14, 1, 1, 29, 147, 155, 28, 1, 64, 586, 1208, 605, 56, 1, 127, 2133, 7819, 7819, 2133, 127, 1, 1, 262, 7288, 44074, 78190, 44074, 7288, 262, 1, 1, 517, 23893, 227569, 655315, 655039, 227623, 23947, 496, 1, 1044, 76332, 1101420
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Number of entries in row n is 1+floor(C(n,2)/2)-floor(C(n-2,2)/2).
Sum of entries in row n is A001710(n) for n>=2.
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REFERENCES
| R. P. Stanley, Binomial posets, Moebius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.
J. Shareshian and M. L. Wachs, q-Eulerian polynomials: excedance number and major index, Electronic Research Announcements of the Amer. Math. Soc., 13 (2007), 33-45.
S. Tanimoto, A study of Eulerian numbers for permutations in the alternating group, Integers, Electronic J. of Combinatorial Number Theory, 6 (2006), #A31.
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FORMULA
| In the Shareshian and Wachs reference (p. 35) a q-analog of the exponential g.f. of the Eulerian polynomials is given for the joint distribution of (inv, des) (see also the Stanley reference). The Maple program given below makes use of this function by considering its even part.
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EXAMPLE
| T(4,2)=5 because we have 4132, 2143, 4213, 2431 and 3241.
Triangle starts:
1;
1;
1,2;
1,5,5,1;
1,14,30,14,1;
1,29,147,155,28;
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MAPLE
| for n to 11 do qbr := proc (m) options operator, arrow; sum(q^i, i = 0 .. m-1) end proc; qfac := proc (m) options operator, arrow; product(qbr(j), j = 1 .. m) end proc; Exp := proc (z) options operator, arrow; sum(q^binomial(m, 2)*z^m/qfac(m), m = 0 .. 19) end proc; g := (1-t)/(Exp(z*(t-1))-t); gser := simplify(series(g, z = 0, 17)); a[n] := simplify(qfac(n)*coeff(gser, z, n)); b[n] := (a[n]+subs(q = -q, a[n]))*1/2; P[n] := sort(subs(q = 1, b[n])) end do; for n to 11 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*binomial(n, 2))-floor((1/2)*binomial(n-2, 2))) end do; # yields sequence in triangular form
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CROSSREFS
| A001710, A145883
Sequence in context: A124660 A141485 A005605 * A111785 A021468 A033282
Adjacent sequences: A145879 A145880 A145881 * A145883 A145884 A145885
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 11 2008
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