A039622 Number of n X n Young tableaux.

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

Offset: 0

Author

Floor van Lamoen

Keywords

• nonn
• nice
• easy

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

Examples

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

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..30
• 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]."]
• Joerg Arndt, The a(3)=42 3 X 3 Young tableaux
• 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, arXiv preprint arXiv:1201.4478 [math.NT], 2012.
• 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 video (2016)
• Curtis Greene and Brady Haran, Shapes and Hook Numbers (extra footage) (2016)
• Zachary Hamaker and Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.
• Alejandro H. Morales, I. Pak, and G. Panova, Why is pi < 2 phi?, Preprint, 2016; The American Mathematical Monthly, Volume 125, 2018 - Issue 8.
• 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.

Crossrefs

Main diagonal of A060854.

Also a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).

Cf. A000178, A000984, A079478, A088020, A107252, A107254, A127223, A153452.

Programs

Magma

A039622:= func< n | n eq 0 select 1 else Factorial(n^2)*(&*[Factorial(j)/Factorial(n+j): j in [0..n-1]]) >;
[A039622(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

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, 12}] (* Jean-François Alcover, Dec 06 2011, after Pari *)

PARI

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

Sage

def A039622(n): return factorial(n^2)*product( factorial(j)/factorial(n+j) for j in (0..n-1))
[A039622(n) for n in (0..12)] # G. C. Greubel, Apr 21 2021

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

From Peter Luschny, May 20 2019: (Start)

a(n) = (G(1+n)*G(2+n)^(2-n)*(n^2)!*(G(3+n)/Gamma(2+n))^(n-1))/(G(1+2*n)*n!) where G(x) is the Barnes G function.

a(n) = A127223(n) / A107252(n). (End)

a(n) = (Gamma(n^2 +1)/Gamma(n+1))*(G(n+1)*G(n+2)/G(2*n+1)), where G(n) is the Barnes G-function. - G. C. Greubel, Apr 21 2021

a(n+2) = (n+2) * A060856(n+1) for n >= 0. - Tom Copeland, May 30 2022