A219311 -id:A219311 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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.

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

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

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

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

Citation corrected by Matthew J. Samuel, Feb 01 2011

A218293 Number of standard Young tableaux with shapes corresponding to partitions into distinct parts.

Original entry on oeis.org

1, 1, 1, 3, 4, 10, 31, 70, 190, 561, 2191, 6226, 22683, 74152, 283349, 1211354, 4572672, 18844177, 77585825, 327472752, 1418056071, 7083303437, 31251988918, 153456264178, 723293387594, 3596567095155, 17360616601051, 89955643932801, 486526881887485, 2551613423040841, 14029592127656040, 76756835252971657, 428044848852530252

Offset: 0

Joerg Arndt, Oct 25 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..120
Wikipedia, Young tableau

Cf. A000085 (standard Young tableaux for all shapes).
Diagonal of A219272, row sums of A219274, A219311. - Alois P. Heinz, Nov 17 2012
Cf. A225121 (tableaux with shapes corresponding to partitions into distinct parts with minimal difference 2).

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) local s; s:=i*(i+1)/2;
      `if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 08 2012

h[l_List] := Module[{n=Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j + Sum[ If[ l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_List] := Module[{s=i*(i+1)/2}, If[n == s, h[Join[l, Table[i-j, {j, 0, i-1}]]], If[n > s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Append[l, i]]]]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

A219320 Number of standard Young tableaux for partitions of n into exactly 7 distinct parts.

Original entry on oeis.org

48608795688960, 295284192952320, 2741894304901440, 18535141513347030, 134524383564933720, 851007098153745060, 5822391651578231460, 37395948352954386420, 238518115727229867660, 1501480486903096567740, 9413700760748972005500, 58167406634979463024710

Offset: 28

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 28..100

Column k=7 of A219311.

Cf. A218293.

A047171 Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= (n-1)/2.

Original entry on oeis.org

0, 0, 0, 2, 3, 9, 14, 34, 55, 125, 209, 461, 791, 1715, 3002, 6434, 11439, 24309, 43757, 92377, 167959, 352715, 646645, 1352077, 2496143, 5200299, 9657699, 20058299, 37442159, 77558759, 145422674, 300540194, 565722719, 1166803109, 2203961429, 4537567649

Offset: 0

Clark Kimberling

nonn

For n>=1 the number of standard Young tableaux with shapes corresponding to partitions into two distinct parts. - Joerg Arndt, Oct 25 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=2 of A219311. - Alois P. Heinz, Nov 17 2012

[0] cat [Binomial(n, Floor((n-1)/2))-1: n in [1..40]]; // Vincenzo Librandi, Jul 03 2015

a:= n-> binomial(n, iquo(n-1,2))-1:
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 17 2012

a[n_] := Binomial[n, Floor[(n-1)/2]]-1; a[0] = 0; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 03 2015 *)

a(n) = A037952(n) - 1. Proof by Ira Gessel: Write down the number of such subsets with k elements <= (n-1)/2 as a product of two binomial coefficients, then evaluate the sum using Vandermonde's theorem.

A219316 Number of standard Young tableaux for partitions of n into exactly 3 distinct parts.

Original entry on oeis.org

16, 35, 134, 435, 1213, 3454, 10484, 28249, 80302, 231895, 638406, 1798515, 5170279, 14361074, 40675562, 116701060, 327587324, 931854890, 2678822398, 7577813175, 21658478151, 62401989636, 177658786252, 509822342794, 1472491312385, 4213745453731, 12134760359950

Offset: 6

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..250

Column k=3 of A219311.

Cf. A218293.

a(n) ~ 3^(n+3/2) / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2014

A219317 Number of standard Young tableaux for partitions of n into exactly 4 distinct parts.

Original entry on oeis.org

768, 2310, 11407, 44187, 200044, 680160, 2769674, 9826918, 38483206, 135059866, 515249581, 1829452107, 6941537898, 24678730371, 92755537994, 333149285650, 1252530682570, 4513808634840, 16936935284163, 61508180909442, 231189178986445, 843098892380280

Offset: 10

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..200

Column k=4 of A219311.

Cf. A218293.

a(n) ~ 2^(2*n+5) / (Pi*n^3). - Vaclav Kotesovec, Sep 13 2014

A219318 Number of standard Young tableaux for partitions of n into exactly 5 distinct parts.

Original entry on oeis.org

292864, 1153152, 7194434, 33888582, 177959434, 861962968, 4036054898, 18519351642, 85808400115, 389017226948, 1778061013340, 7967135309510, 35973133665285, 161398383117645, 726152765571840, 3256479005867430, 14629885404315411, 65641088599945380

Offset: 15

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 15..100

Column k=5 of A219311.

Cf. A218293.

A219319 Number of standard Young tableaux for partitions of n into exactly 6 distinct parts.

Original entry on oeis.org

1100742656, 5462865408, 42035926724, 238839304110, 1477773782690, 8119282473120, 49406279584740, 259405071568305, 1468158383705685, 7798557001165665, 42744495396935010, 224697430361576340, 1226112009886575180, 6397760480647576200, 34422065224987469772

Offset: 21

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 21..100

Column k=6 of A219311.

Cf. A218293.

A219321 Number of standard Young tableaux for partitions of n into exactly 8 distinct parts.

Original entry on oeis.org

29258366996258488320, 212593716124699852800, 2338077979922915527680, 18541315347775500731880, 156386347073221236234900, 1136852065645214098726260, 8834800018708598317055880, 63128042819798223843289680, 473458147812239316816345390, 3325190851643455231559076330

Offset: 36

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 36..100

Column k=8 of A219311.

Cf. A218293.

A219322 Number of standard Young tableaux for partitions of n into exactly 9 distinct parts.

Original entry on oeis.org

273035280663535522487992320, 2329440559042398325938585600, 29867922172910654180714311680, 274273837545154589560694664960, 2661165012109500556315841832780, 22087643957707583932955480283900, 194567964473214023234175600529800, 1559754564113482833062794519391700

Offset: 45

Alois P. Heinz, Nov 17 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 45..100

Column k=9 of A219311.

Cf. A218293.

cp's OEIS Frontend

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

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A218293 Number of standard Young tableaux with shapes corresponding to partitions into distinct parts.

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A219320 Number of standard Young tableaux for partitions of n into exactly 7 distinct parts.

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A047171 Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= (n-1)/2.

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A219316 Number of standard Young tableaux for partitions of n into exactly 3 distinct parts.

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A219317 Number of standard Young tableaux for partitions of n into exactly 4 distinct parts.

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A219318 Number of standard Young tableaux for partitions of n into exactly 5 distinct parts.

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A219319 Number of standard Young tableaux for partitions of n into exactly 6 distinct parts.

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A219321 Number of standard Young tableaux for partitions of n into exactly 8 distinct parts.

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A219322 Number of standard Young tableaux for partitions of n into exactly 9 distinct parts.

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