This site is supported by donations to The OEIS Foundation.



Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 4500 articles have referenced us, often saying "we would not have discovered this result without the OEIS".

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)



Number of arrangements of 1,2,..,n^2 in an n X n matrix such that each row and each column is increasing.

Factor g_n in formula for 2n-th moment of Riemann zeta function on the critical line. (See Conrey article.) - Michael Somos, Apr 15 2003

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


The problem for a 5 X 5 array was recently posed and solved in the College Mathematics Journal. The solution is in Vol. 30 (1999), no. 5, pp. 410-411.

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


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

P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702, 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.

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.

Index entries for sequences related to Young tableaux.


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


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


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

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


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 *)


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


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

Sequence in context: A193272 A193273 A182192 * A130506 A052078 A069544

Adjacent sequences:  A039619 A039620 A039621 * A039623 A039624 A039625




Floor van Lamoen


References, correction and extension from Stephen G. Penrice (spenrice(AT)ets.org), Jun 15 2000



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 25 01:19 EST 2015. Contains 264374 sequences.