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A039622 Number of n X n Young tableaux. 9
1, 1, 2, 42, 24024, 701149020, 1671643033734960, 475073684264389879228560, 22081374992701950398847674830857600, 220381378415074546123953914908618547085974856000, 599868742615440724911356453304513631101279740967209774643120000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of arrangements of 1,2,..,n^2 in an n X n array such that each row and each column is increasing. The problem for a 5 X 5 array was recently posed and solved in the College Mathematics Journal. See the links.

This is the factor g_n that appears in a conjectured formula for 2n-th moment of the Riemann zeta function on the critical line. (See Conrey articles.) - Michael Somos, Apr 15 2003 [Comment revised by N. J. A. Sloane, Jun 21 2016]

Number of linear extensions of the n X n lattice. - Mitch Harris, Dec 27 2005

REFERENCES

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 284.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20

P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]

J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 349.

J. B. Conrey, Review of H. Iwaniec, "Lectures on the Riemann Zeta Function" (AMS, 2014), Bull. Amer. Math. Soc., 53 (No. 3, 2016), 507-512.

P.-O. Dehaye, Combinatorics of the lower order terms in the moment conjectures: the Riemann zeta function

J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-324.

Curtis Greene and Brady Haran, Shapes and Hook Numbers - Numberphile (2016)

Curtis Greene and Brady Haran, Shapes and Hook Numbers (extra footage) (2016)

Alan H. Rapoport (proposer), Solution to Problem 639: A Square Young Tableau, College Mathematics Journal, Vol. 30 (1999), no. 5, pp. 410-411.

Index entries for sequences related to Young tableaux.

FORMULA

a(n) = (n^2)! / (product k=1, ..., 2n-1 k^(n - |n-k|)).

a(n) = 0!*1!*..*(k-1)! *(k*n)! / ( n!*(n+1)!*..*(n+k-1)! ) for k=n.

a(n) = A088020(n)/A107254(n) = A088020(n)*A000984(n)/A079478(n). - Henry Bottomley, May 14 2005

a(n) = A153452(prime(n)^n). - Naohiro Nomoto, Jan 01 2009

a(n) ~ sqrt(Pi) * n^(n^2+11/12) * exp(n^2/2+1/12) / (A * 2^(2*n^2-7/12)), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015

EXAMPLE

Using the hook length formula, a(4) = (16)!/(7*6^2*5^3*4^4*3^3*2^2) = 24024.

MAPLE

a:= n-> (n^2)! *mul(k!/(n+k)!, k=0..n-1):

seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2012

MATHEMATICA

a[n_] := (n^2)!*Product[ k!/(n + k)!, {k, 0, n - 1}]; Table[ a[n], {n, 0, 9}] (* Jean-Fran├žois Alcover, Dec 06 2011, after Pari *)

PROG

(PARI) a(n)=(n^2)!*prod(k=0, n-1, k!/(n+k)!)

CROSSREFS

Main diagonal of A060854. Also a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).

Sequence in context: A193272 A193273 A182192 * A130506 A273399 A052078

Adjacent sequences:  A039619 A039620 A039621 * A039623 A039624 A039625

KEYWORD

nonn,nice,easy

AUTHOR

Floor van Lamoen

STATUS

approved

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Last modified December 9 14:38 EST 2016. Contains 278971 sequences.