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A002538
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Second-order Eulerian numbers <<n+1,n-1>>
(Formerly M4548 N1932)
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8
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1, 8, 58, 444, 3708, 33984, 341136, 3733920, 44339040, 568356480, 7827719040, 115336085760, 1810992556800, 30196376985600, 532953524275200, 9927928075161600
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Second-order eulerian numbers <<n,k>> count permutations of the multiset {1,1,2,2,...,n,n} with k ascents and the restriction that for any m <= n, all numbers between the two copies of m are less than m.
a(n) = number of edges in the Hasse diagram for the Bruhat order on permutations of [n+1]. - David Callan (callan(AT)stat.wisc.edu), Sep 03 2005
Proof. As explained on page 1 of the Stanley link, edges in the Hasse diagram of the (strong) Bruhat order on S_n are associated with pairs (pi,(i,j)) with pi in S_n and 1 <= i < j <= n, such that pi_i < pi_j and each entry of pi lying between pi_i and pi_j in POSITION does not lie between pi_i and pi_j in VALUE.
For example, pi = (3, 5, 1, 2, 4) gives edges for the (i,j) pairs (1,2), (1,5), (3,4), (4,5) but not, e.g., for (i,j) = (3,5) because 2 lies between pi_3=1 and pi_5=4 both in position and in value.
Let us count edges for a given pair (i,j). Consider the j-i+1 entries pi_i, pi_(i+1),...,pi_j. There are (j-i+1)! possible orderings for their values and (i,j) contributes an edge <=> the values of pi_i, pi_j are adjacent in this ordering with pi_i < pi_j.
There are (j-i)! such orderings (just coalesce the items pi_i, pi_j into a single item). The net result is that (i,j) contributes an edge 1/(j-i+1) of the time. So the total number of edges in the Hasse diagram is Sum_{1 <= i < j <= n} n!/(j-i+1) = (n+1)!(H_(n+1) - 2) + n! where H_n = 1+1/2+1/3+ ... +1/n is the harmonic sum. QED - David Callan (callan(AT)stat.wisc.edu), Mar 07 2006
Number of reentrant corners along the lower contours of all deco polyominoes of height n+2. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. a(n)=Sum(k*A121579(n+2,k), k>0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2006
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REFERENCES
| E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33. (See Table 1.)
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
R. P. Stanley, A survey of the Bruhat order of the symmetric group
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FORMULA
| Sum_{k=1..n} k*|Stirling1(n+2, k+2)|. E.g.f.: (x+2*ln(1-x))/(x-1)^3. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 15 2003
With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at -1. - Ralf Stephan, Apr 16 2004
a(n)=(n+2)*a(n-1) + n*n!, n>=1, a(0):=0.
a(n)=(n+2)!HarmonicSum[n+2]+(n+1)!-2(n+2)! where HarmonicSum[n]=1+1/2+1/3+...+1/n. - David Callan (callan(AT)stat.wisc.edu), Mar 07 2006
a(n)=(n+1)!*((n+2)*h(n+2)-2*n-3) where h(n)=sum(1/k,k=1..n) [From Gary Detlefs (gdetlefs(AT)aol.com), Mar 25 2011]
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MATHEMATICA
| Table[(-1)^(n + 1)* Sum[(-1)^(n - k) k (-1)^(n - k) StirlingS1[n + 3, k + 3], {k, 0, n}], {n, 1, 16}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
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CROSSREFS
| Second diagonal of A008517 and second column of A112007.
Cf. A121579.
Sequence in context: A126529 A039759 A047867 * A112424 A190977 A186362
Adjacent sequences: A002535 A002536 A002537 * A002539 A002540 A002541
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Mira Bernstein, Robert G. Wilson v (rgwv(AT)rgwv.com)
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