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A003121
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Strict sense ballot numbers: n candidates, k-th candidate gets k votes.
(Formerly M2048)
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6
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1, 1, 2, 12, 286, 33592, 23178480, 108995910720, 3973186258569120, 1257987096462161167200, 3830793890438041335187545600, 123051391839834932169117010215648000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Also, number of even minus number of odd extensions of truncated n-1 by n grid diagram.
Also, a(n) is the degree of the spinor variety, the complex projective variety SO(2n+1)/U(n). See Hiller's paper. - Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
Also, number of ways of placing 1,...,n(n+1)/2 in a triangular array such that both rows and columns are increasing, i.e., the number of shifted standard Young tableaux of shape (n, n - 1, ..., 1).
E.g. a(3)=2 as we can write:
1
23
456
or
1
24
356
(or transpose these to have shifted tableaux). - Jon Perry, Jun 13 2003, edited by Joel Brewster Lewis, Aug 27 2011.
Also, the number of symbolic sequences on the n symbols 0,1, ..., n-1. A symbolic sequence is a sequence that has n occurrences of 0, n-1 occurrences of 1, ..., one occurrence of n-1 and satisfies the condition that between any two consecutive occurrences of the symbol i it has exactly one occurrence of the symbol i+1. For example, the two symbolic sequences on 3 symbols are 012010 and 010210. The Shapiro-Shapiro paper shows how such sequences arise in the study of the arrangement of the real roots of a polynomial and its derivatives. There is a natural bijection between symbolic sequences and the triangular arrays described above. - Peter Bala, Jul 18 2007
a(n) also appears to be the number of chains from w_0 down to 1 in a certain suborder of the strong Bruhat order on S_n: we let v cover (ij)*v only if i,j straddle the leftmost descent in v. For n=3 and drawing that descent with a |, this order is 3|21 > 23|1 > 13|2 & 2|13 > 123, with two maximal chains. [From Allen Knutson (allenk(AT)math.cornell.edu), Oct 13 2008]
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REFERENCES
| D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236-260.
H. Hiller. Combinatorics and intersection of Schubert varieties. Comment. Math. Helv. 57 (1982), 41-59.
G. Kreweras, Sur un probleme de scrutin a plus de deux candidats, Publications de l'Institut de Statistique de l'Universit\'{e} de Paris, 26 (1981), 69-87.
F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.
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LINKS
| D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236-260.
B. Shapiro and M. Shapiro, A few riddles behind Rolle's theorem
R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.
Dennis White, Sign-balanced posets
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FORMULA
| C(n+1, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!)
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PROG
| (PARI) a(n)=((n*n+n)/2)!*prod(i=1, n, (i-1)!/(2*i-1)!)
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CROSSREFS
| Cf. A005118, A018241, A007724, A004065, A131811.
Sequence in context: A012444 A012754 A083568 * A176037 A057170 A200564
Adjacent sequences: A003118 A003119 A003120 * A003122 A003123 A003124
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KEYWORD
| nonn,nice,easy
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AUTHOR
| C. L. Mallows (colinm(AT)research.avayalabs.com)
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EXTENSIONS
| More terms from Michael Somos . Additional references from Frank Ruskey (ruskey(AT)cs.uvic.ca)
Formula corrected by Eric Rowland (erowland(AT)tulane.edu), Jun 18 2010
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