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Search: id:A003121
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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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+0 7
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| 1, 1, 2, 12, 286, 33592, 23178480, 108995910720, 3973186258569120, 1257987096462161167200, 3830793890438041335187545600, 123051391839834932169117010215648000
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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 in a triangular array such that both rows and columns are increasing. - Jon Perry (perry(AT)globalnet.co.uk), Jun 13 2003
E.g. a(3)=2 as we can write:
1
23
456
or
1
24
356
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 (pbala(AT)toucansurf.com), Jul 18 2007
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REFERENCES
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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.
R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.
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LINKS
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D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236-260.
F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.
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
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C(n, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!)
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PROGRAM
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(PARI) a(n)=((n*n+n)/2)!*prod(i=1, n, (i-1)!/(2*i-1)!)
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CROSSREFS
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Cf. A005118, A018241, A007724, A004065.
Cf. A131811.
Adjacent sequences: A003118 A003119 A003120 this_sequence A003122 A003123 A003124
Sequence in context: A012444 A012754 A083568 this_sequence A057170 A008338 A000178
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KEYWORD
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nonn,nice,easy
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AUTHOR
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C. L. Mallows (colinm(AT)research.avayalabs.com)
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EXTENSIONS
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More terms from Michael Somos . Additional references from Frank Ruskey (fruskey(AT)cs.uvic.ca)
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