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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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14
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1, 1, 1, 2, 12, 286, 33592, 23178480, 108995910720, 3973186258569120, 1257987096462161167200, 3830793890438041335187545600, 123051391839834932169117010215648000, 45367448380314462649742951646437285434233600, 207515126854334868747300581954534054343817468395494400
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OFFSET
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0,4
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COMMENTS
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Also, number of even minus number of odd extensions of truncated (n-1) X 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 1
2 3 or 2 4
4 5 6 3 5 6
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. - Allen Knutson (allenk(AT)math.cornell.edu), Oct 13 2008
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
6
4 8
2 5 9
1 3 7 10
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
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
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
Number of maximal chains of length binomial(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
The normalized volume of the subpolytope of the Birkhoff polytope obtained by taking the convex hull of all n X n permutation matrices corresponding to permutations that avoid the patterns 132 and 312. - Robert Davis, Dec 04 2016
The number of maximal chains in the lattice of permutations which avoid both the patterns 132 and 312, as a sublattice of the weak order on the symmetric group. For example, there are exactly 12 maximal chains in the sublattice for the weak order on the symmetric group on 5 elements.
The number of words in the commutation class of the c-sorting word of the longest permutation w_0 in the symmetric group for the Coxeter element c = s_1 s_2 s_3 s_4 s_5 ... . For example, the c-sorting word of w_0 for s_1 s_2 s_3 s_4 is the reduced word s_1 s_2 s_3 s_4 | s_1 s_2 s_3 | s_1 s_2 | s_1, and there are exactly 12 words in its commutation class.
The number of maximal chains in the lattice of c-singletons for the symmetric group, for the Coxeter element c = s_1 s_2 s_3 s_4 s_5 ... . For example, there are exactly 12 maximal chains in the lattice of c-singletons for c = s_1 s_2 s_3 s_4. (End)
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REFERENCES
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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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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William T. Dugan, Maura Hegarty, Alejandro H. Morales, and Annie Raymond, Generalized Pitman-Stanley flow polytopes, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #80.
N. Reading, Cambrian lattices, arXiv:math/0402086 [math.CO], 2004-2005; Adv. Math. 205 (2006), no. 2, 313-353.
Dennis White, Sign-balanced posets, Journal of Combinatorial Theory, Series A, Volume 95, Issue 1, July 2001, Pages 1-38.
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FORMULA
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a(n) = binomial(n+1, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!).
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
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EXAMPLE
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The a(4) = 12 ways to fill a triangle with the numbers 0 through 9:
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5 6 6 5
3 7 3 7 2 7 2 7
1 4 8 1 4 8 1 4 8 1 4 8
0 2 6 9 0 2 5 9 0 3 5 9 0 3 6 9
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5 3 3 4
3 6 2 6 2 7 3 7
1 4 8 1 5 8 1 5 8 1 5 8
0 2 7 9 0 4 7 9 0 4 6 9 0 2 6 9
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4 4 5 4
2 6 2 7 2 6 3 6
1 5 8 1 5 8 1 4 8 1 5 8
0 3 7 9 0 3 6 9 0 3 7 9 0 2 7 9
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(End)
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MAPLE
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f:= n-> ((n*n+n)/2)!*mul((i-1)!/(2*i-1)!, i=1..n); seq(f(n), n=0..20);
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MATHEMATICA
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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 *)
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PROG
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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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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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