A005118 Number of simple allowable sequences on 1..n containing the permutation 12...n.

1, 1, 1, 2, 16, 768, 292864, 1100742656, 48608795688960, 29258366996258488320, 273035280663535522487992320, 44261486084874072183645699204710400, 138018895500079485095943559213817088756940800

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

For n >= 2 by the hook length formula a(n) is also the number of Young tableaux of size 1+2+...+(n-1) = n*(n-1)/2 that correspond to the partition (1,2,...n-1), i.e., triangular Young tableaux. For example, for n=5 the shape of the tableau is xxxx / xxx / xx / x. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001

Also, a(n) is the degree of the symplectic Grassmannian, the projective variety of all maximal isotropic subspaces in a complex vector space of dimension 2n-2 with a symplectic form. See Hiller's paper. - Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002

Also, for n >= 2, a(n) is the number of maximal chains in the poset of Dyck paths ordered by inclusion. - Jennifer Woodcock (Jennifer.Woodcock(AT)ugdsb.on.ca), May 21 2008

a(n) is the number of minimal decompositions of the "flip" permutation n(n-1)..21 in terms of the n-1 standard Coxeter generators (i i+1) ("reduced decompositions", cf. Stanley). As such, it is also the number of positive n-strand braid words representing the Garside braid Delta(n) (the half-turn) (cf. Epstein's book, lemma 9.1.14). - Maxime Bourrigan, Apr 04 2011

For n >= 1, the normalized volume of the subpolytope of the Birkhoff polytope obtained by taking the convex hull of all (2n)x(2n) permutation matrices corresponding to alternating permutations that also avoid the pattern 123. - Robert Davis, Dec 04 2016

References

• D. B. A. Epstein with J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson and W. P. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp.
• J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
• 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).

Links

• T. D. Noe, Table of n, a(n) for n = 0..40
• Omer Angel, Alexander E. Holroyd, Dan Romik, and Balint Virag, Random Sorting Networks, arXiv preprint arXiv:0609538 [math.PR], 2006.
• Joerg Arndt, The a(4)=16 Young tableaux of shape [3, 2, 1].
• Sara C. Billey and Peter R. W. McNamara, The contributions of Stanley to the fabric of symmetric and quasisymmetric functions, arXiv preprint, 2015.
• Tobias Boege, Alessio D'Alì, Thomas Kahle, Bernd Sturmfels, The Geometry of Gaussoids, arXiv:1710.07175 [math.CO], 2017.
• R. Davis and B. Sagan, Pattern-Avoiding Polytopes, 2016
• FindStat - Combinatorial Statistic Finder, The number of ways to write a permutation as a minimal length product of simple transpositions
• M. J. Hay, J. Schiff, and N. J. Fisch, Maximal energy extraction under discrete diffusive exchange, arXiv preprint arXiv:1508.03499 [physics.plasm-ph], 2015.
• H. Hiller, Combinatorics and intersection of Schubert varieties, Comment. Math. Helv. 57 (1982), 41-59.
• D. Kim, Finding k Shortest Paths in Cayley Graphs of Finite Groups, Graphs and Combinatorics 40, 120 (2024). See Formula at p. 13.
• 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. [Annotated scanned copy]
• Joshua Maglione and Christopher Voll, Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration, arXiv:2410.08075 [math.CO], 2024. See pp. 34, 39.
• R. P. Stanley, A combinatorial miscellany
• R. P. Stanley, Ordering events in Minkowski space, arXiv:math/0501256 [math.CO], 2005.
• R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5 (1984), 359-372.

Crossrefs

Cf. A003121, A018241, A057863, A246865, A289778.

Programs

Maple

A005118 := proc(n) local i; binomial(n,2)!/product( (2*i+1)^(n-i-1), i=0..n-2 ); end;

Mathematica

Table[Binomial[n, 2]!/Product[(2*i + 1)^(n - i - 1), {i, 0, n - 2}], {n, 0, 10}] (* T. D. Noe, May 29 2012 *)

Formula

a(n) = C(n, 2)!/(1^{n-1} * 3^{n-2} *...* (2n-3)^1 ).

a(n) = (n*(n-1)/2)!/A057863(n-1) (n>=1). - Emeric Deutsch, May 21 2004

a(n) = A153452(A002110(n-1)). - Naohiro Nomoto, Jan 01 2009

From Alois P. Heinz, Nov 18 2012: (Start)

a(n+1) = A219272(A000217(n),n) = A219274(A000217(n),n) = A219311(A000217(n),n).

a(n) = A193536(n,A000217(n-1)) = A193629(n,A000217(n-1)). (End)

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

Extensions

Citation corrected by Matthew J. Samuel, Feb 01 2011