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 A003121 Strict sense ballot numbers: n candidates, k-th candidate gets k votes. (Formerly M2048) 12

%I M2048

%S 1,1,1,2,12,286,33592,23178480,108995910720,3973186258569120,

%T 1257987096462161167200,3830793890438041335187545600,

%U 123051391839834932169117010215648000,45367448380314462649742951646437285434233600,207515126854334868747300581954534054343817468395494400

%N Strict sense ballot numbers: n candidates, k-th candidate gets k votes.

%C Also, number of even minus number of odd extensions of truncated n-1 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, ..., 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

%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 3|21 > 23|1 > 13|2 & 2|13 > 123, with two maximal chains. - 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 (rc-graphs) 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+j-1,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, (n-1) 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

%C Regarding the comment dated Mar 25 2012, it is assumed that each row is in increasing order, as in the example dated Jul 12 2012. How many row-interlacing triangles are there without that restriction? - _Clark Kimberling_, Dec 02 2014

%C Number of maximal chains of length C(n+1,2) in the Tamari lattice T_{n+1}. For n=2 there is 1 maximal chain of length 3 in the Tamari lattice T_3. - _Alois P. Heinz_, Dec 04 2015

%C The normalized volume of the subpolytope of the Birkhoff polytope obtained by taking the convex hull of all nxn permutation matrices corresponding to permutations that avoid the patterns 132 and 312. - _Robert Davis_, Dec 04 2016

%D G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane, <a href="/A003121/b003121.txt">Table of n, a(n) for n = 0..30</a>

%H E. Aas and S. Linusson, <a href="http://www.diva-portal.org/smash/get/diva2:768232/FULLTEXT01.pdf">Continuous multiline queues and TASEP</a>, 2014

%H D. E. Barton and C. L. Mallows, <a href="http://dx.doi.org/10.1214/aoms/1177700286">Some aspects of the random sequence</a>, Ann. Math. Statist. 36 (1965) 236-260.

%H R. Davis and B. Sagan, <a href="https://arxiv.org/abs/1609.01782">Pattern-Avoiding Polytopes</a>, arXiv:1609.01782 [math.CO], 2016.

%H H. Hiller, <a href="http://dx.doi.org/10.5169/seals-43873">Combinatorics and intersection of Schubert varieties</a>, Comment. Math. Helv. 57 (1982), 41-59. [Broken link]

%H G. Kreweras, <a href="/A005118/a005118.pdf">Sur un problème de scrutin à plus de deux candidats</a>, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87. [Annotated scanned copy]

%H Colin Mallows, <a href="/A003121/a003121.pdf">Letter to N. J. A. Sloane, date unknown</a>.

%H F. Ruskey, <a href="http://dx.doi.org/10.1016/0095-8956(92)90067-8">Generating linear extensions of posets by transpositions</a>, J. Combin. Theory, B 54 (1992), 77-101.

%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), 787-795.

%H George Story, <a href="http://www.rose-hulman.edu/mathjournal/archives/2013/vol14-n1/paper12/v14n1-12pd.pdf">Counting Maximal Chains in Weighted Voting Posets</a>, Rose-Hulman Undergraduate Mathematics Journal, Vol. 14, No. 1, 2013.

%H R. M. Thrall, <a href="http://dx.doi.org/10.1307/mmj/1028989731">A combinatorial problem</a>, Michigan Math. J. 1, (1952), 81-88.

%H Dennis White, <a href="http://dx.doi.org/10.1006/jcta.2000.3146">Sign-balanced posets</a>, Journal of Combinatorial Theory, Series A, Volume 95, Issue 1, July 2001, Pages 1-38.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tamari_lattice">Tamari lattice</a>

%F a(n) = C(n+1, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!).

%F a(n) ~ sqrt(Pi) * exp(n^2/4+n/2+7/24) * n^(n^2/2+n/2+23/24) / (A^(1/2) * 2^(3*n^2/2+n+5/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Nov 13 2014

%e The a(4) = 12 ways to fill a triangle with the numbers 0 through 9:

%e .

%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 .

%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 .

%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 .

%e - _R. H. Hardin_, Jul 06 2012

%p f:= n-> ((n*n+n)/2)!*mul((i-1)!/(2*i-1)!, 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,(i-1)!/(2*i-1)!)

%Y Cf. A005118, A018241, A007724, A004065, A131811, A064049, A064050.

%Y A213457 is also closely related.

%Y Cf. A000108, A027686.

%K nonn,nice,easy,changed

%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

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