%I M2048
%S 1,1,1,2,12,286,33592,23178480,108995910720,3973186258569120,
%T 1257987096462161167200,3830793890438041335187545600,
%U 123051391839834932169117010215648000,45367448380314462649742951646437285434233600
%N Strict sense ballot numbers: n candidates, kth candidate gets k votes.
%C Also, number of even minus number of odd extensions of truncated n1 by n grid diagram.
%C 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 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).
%C E.g. a(3) = 2 as we can write:
%C 1 1
%C 2 3 or 2 4
%C 4 5 6 3 5 6
%C (or transpose these to have shifted tableaux).  _Jon Perry_, Jun 13 2003, edited by _Joel B. Lewis_, Aug 27 2011.
%C Also, the number of symbolic sequences on the n symbols 0,1, ..., n1. A symbolic sequence is a sequence that has n occurrences of 0, n1 occurrences of 1, ..., one occurrence of n1 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 ShapiroShapiro 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
%C 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 321 > 231 > 132 & 213 > 123, with two maximal chains. [From Allen Knutson (allenk(AT)math.cornell.edu), Oct 13 2008]
%C Number of ways to arrange the numbers 1,2,...,n(n+1)/2 in a triangle so that the rows interlace; e.g. one of the 12 triangles counted by a(4) is
%C ......6
%C ....4...8
%C ..2...5...9
%C 1...3...7...10
%C [_Clark Kimberling_, Mar 25 2012]
%C Also, the number of maps from n x n pipe dreams (rcgraphs) to words of adjacent transpositions in S_n that send a crossing of pipes x and y in square (i,j) to the transposition (i+j1,i+j) swapping x and y. Taking the pictorial image of a permutation as a wiring diagram, these are maps from pipe dreams to wiring diagrams that send crossings of pipes to crossings of similarly labeled wires.  _Cameron Marcott_, Nov 26 2012
%C Number of words of length T(n)=n*(n+1)/2 with n 1's, (n1) 2's, ..., and 1 n such that counting the numbers from left to right we always have 1>2>3>...>n. The 12 words for n=4 are 1111222334, 1111223234, 1112122334, 1112123234, 1112212334, 1112213234, 1112231234, 1121122334, 1121123234, 1121212334, 1121213234 and 1121231234.  _Jon Perry_, Jan 27 2013
%D D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236260.
%D H. Hiller. Combinatorics and intersection of Schubert varieties. Comment. Math. Helv. 57 (1982), 4159.
%D 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), 6987.
%D F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77101.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D George Story, Counting Maximal Chains in Weighted Voting Posets, RoseHulman Undergraduate Mathematics Journal, Vol. 14, No. 1, 2013.
%D R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 8188.
%H 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) 236260.
%H B. Shapiro and M. Shapiro, <a href="http://arXiv.org/abs/math.CA/0302215">A few riddles behind Rolle's theorem</a>, Amer. Math. Monthly, 119 (2012), 787795.
%H 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), 8188.
%H Dennis White, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.3034">Signbalanced posets</a>
%H N. J. A. Sloane, <a href="/A003121/b003121.txt">Table of n, a(n) for n = 0..30</a>
%F a(n) = C(n+1, 2)!*(1!*2!*...*(n1)!)/(1!*3!*...*(2n1)!).
%e The a(4) = 12 ways to fill a triangle with the numbers 0 through 9:
%e .....5........6........6........5..
%e ....3.7......3.7......2.7......2.7..
%e ...1.4.8....1.4.8....1.4.8....1.4.8..
%e ..0.2.6.9..0.2.5.9..0.3.5.9..0.3.6.9..
%e .....5........3........3........4..
%e ....3.6......2.6......2.7......3.7..
%e ...1.4.8....1.5.8....1.5.8....1.5.8..
%e ..0.2.7.9..0.4.7.9..0.4.6.9..0.2.6.9..
%e .....4........4........5........4..
%e ....2.6......2.7......2.6......3.6..
%e ...1.5.8....1.5.8....1.4.8....1.5.8..
%e ..0.3.7.9..0.3.6.9..0.3.7.9..0.2.7.9..
%e  _R. H. Hardin_, Jul 06 2012
%p f:= n> ((n*n+n)/2)!*mul((i1)!/(2*i1)!, i=1..n); seq (f(n), n=0..20);
%t f[n_] := (n (n + 1)/2)! Product[(i  1)!/(2 i  1)!, {i, n}]; Array[f, 14, 0] (* _Robert G. Wilson v_, May 14 2013 *)
%o (PARI) a(n)=((n*n+n)/2)!*prod(i=1,n,(i1)!/(2*i1)!)
%Y Cf. A005118, A018241, A007724, A004065, A131811, A064049, A064050.
%Y A213457 is also closely related.
%Y Cf. A000108.
%K nonn,nice,easy
%O 0,4
%A _Colin Mallows_
%E More terms from _Michael Somos_.
%E Additional references from _Frank Ruskey_
%E Formula corrected by _Eric Rowland_, Jun 18 2010
