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A039622 Number of n X n Young tableaux. 13
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 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


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..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]."]

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.


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


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, 12}] (* Jean-Fran├žois Alcover, Dec 06 2011, after Pari *)


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


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


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


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.

Sequence in context: A193273 A182192 A330229 * A130506 A273399 A052078

Adjacent sequences:  A039619 A039620 A039621 * A039623 A039624 A039625




Floor van Lamoen



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Last modified September 28 20:34 EDT 2022. Contains 357081 sequences. (Running on oeis4.)