%I A003121 M2048
%S A003121 1,1,2,12,286,33592,23178480,108995910720,3973186258569120,
%T A003121 1257987096462161167200,3830793890438041335187545600,
%U A003121 123051391839834932169117010215648000
%N A003121 Strict sense ballot numbers: n candidates, k-th candidate gets k votes.
%C A003121 Also, number of even minus number of odd extensions of truncated n-1 
               by n grid diagram.
%C A003121 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
%C A003121 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
%C A003121 E.g. a(3)=2 as we can write:
%C A003121 1
%C A003121 23
%C A003121 456
%C A003121 or
%C A003121 1
%C A003121 24
%C A003121 356
%C A003121 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
%C A003121 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]
%D A003121 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003121 D. E. Barton and C. L. Mallows, Some aspects of the random sequence, 
               Ann. Math. Statist. 36 (1965) 236-260.
%D A003121 H. Hiller. Combinatorics and intersection of Schubert varieties. Comment. 
               Math. Helv. 57 (1982), 41-59.
%D A003121 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.
%D A003121 R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.
%H A003121 D. E. Barton and C. L. Mallows, <a href="http://projecteuclid.org/DPubS/
               Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/
               1177700286">Some aspects of the random sequence</a>, Ann. Math. Statist. 
               36 (1965) 236-260.
%H A003121 F. Ruskey, <a href="http://www.cs.uvic.ca/~fruskey/Publications/JCTextension.html">
               Generating linear extensions of posets by transpositions</a>, J. 
               Combin. Theory, B 54 (1992), 77-101.
%H A003121 B. Shapiro and M. Shapiro, <a href="http://arXiv.org/abs/math.CA/0302215">
               A few riddles behind Rolle's theorem</a>
%H A003121 R. M. Thrall, <a href="http://projecteuclid.org/Dienst/Repository/1.0/
               Disseminate/euclid.mmj/1028989731/body/pdf">A combinatorial problem</
               a>, Michigan Math. J. 1, (1952), 81-88.
%H A003121 Dennis White, <a href="http://citeseer.ist.psu.edu/cache/papers/cs/24299/
               http:zSzzSzwww.math.umn.eduzSz~whitezSzpaperszSzsignbal.pdf/white00signbalanced.pdf">
               Sign-balanced posets</a>
%F A003121 C(n, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!)
%o A003121 (PARI) a(n)=((n*n+n)/2)!*prod(i=1,n,(i-1)!/(2*i-1)!)
%Y A003121 Cf. A005118, A018241, A007724, A004065.
%Y A003121 Cf. A131811.
%Y A003121 Sequence in context: A012444 A012754 A083568 this_sequence A057170 A008338 
               A000178
%Y A003121 Adjacent sequences: A003118 A003119 A003120 this_sequence A003122 A003123 
               A003124
%K A003121 nonn,nice,easy
%O A003121 1,3
%A A003121 C. L. Mallows (colinm(AT)research.avayalabs.com)
%E A003121 More terms from Michael Somos . Additional references from Frank Ruskey 
               (fruskey(AT)cs.uvic.ca)