

A039622


Number of n X n Young tableaux.


9



1, 1, 2, 42, 24024, 701149020, 1671643033734960, 475073684264389879228560, 22081374992701950398847674830857600, 220381378415074546123953914908618547085974856000, 599868742615440724911356453304513631101279740967209774643120000
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OFFSET

0,3


COMMENTS

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 2nth 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


REFERENCES

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. 410411.
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, 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), 341353. 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. 316324.
Index entries for sequences related to Young tableaux.


FORMULA

a(n) = (n^2)! / (product k=1, ..., 2n1 k^(n  nk)).
a(n) = 0!*1!*..*(k1)! *(k*n)! / ( n!*(n+1)!*..*(n+k1)! ) 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^27/12)), where A = 1.28242712910062263687534256886979... is the GlaisherKinkelin 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..n1):
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}] (* JeanFrançois Alcover, Dec 06 2011, after Pari *)


PROG

(PARI) a(n)=if(n<0, 0, (n^2)!*prod(k=0, n1, k!/(n+k)!))


CROSSREFS

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


KEYWORD

nonn,nice,easy


AUTHOR

Floor van Lamoen


EXTENSIONS

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


STATUS

approved



