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%I #97 Dec 30 2023 12:10:28
%S 1,1,2,42,24024,701149020,1671643033734960,475073684264389879228560,
%T 22081374992701950398847674830857600,
%U 220381378415074546123953914908618547085974856000,599868742615440724911356453304513631101279740967209774643120000
%N Number of n X n Young tableaux.
%C 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.
%C 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]
%C Number of linear extensions of the n X n lattice. - _Mitch Harris_, Dec 27 2005
%D M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 284.
%H Alois P. Heinz, <a href="/A039622/b039622.txt">Table of n, a(n) for n = 0..30</a>
%H P. Aluffi, <a href="http://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, 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]."]
%H Joerg Arndt, <a href="/A039622/a039622.txt">The a(3)=42 3 X 3 Young tableaux</a>
%H J. B. Conrey, <a href="http://www.ams.org/notices/200303/fea-conrey-web.pdf">The Riemann Hypothesis</a>, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 349.
%H J. B. Conrey, <a href="http://dx.doi.org/10.1090/bull/1525">Review of H. Iwaniec, "Lectures on the Riemann Zeta Function" (AMS, 2014)</a>, Bull. Amer. Math. Soc., 53 (No. 3, 2016), 507-512.
%H P.-O. Dehaye, <a href="https://arxiv.org/abs/1201.4478">Combinatorics of the lower order terms in the moment conjectures: the Riemann zeta function</a>, arXiv preprint arXiv:1201.4478 [math.NT], 2012.
%H J. S. Frame, G. de B. Robinson and R. M. Thrall, <a href="http://dx.doi.org/10.4153/CJM-1954-030-1">The hook graphs of a symmetric group</a>, Canad. J. Math. 6 (1954), pp. 316-324.
%H Curtis Greene and Brady Haran, <a href="https://www.youtube.com/watch?v=vgZhrEs4tuk">Shapes and Hook Numbers</a>, Numberphile video (2016)
%H Curtis Greene and Brady Haran, <a href="https://www.youtube.com/watch?v=JiwM5g_RC3c">Shapes and Hook Numbers (extra footage)</a> (2016)
%H Zachary Hamaker and Eric Marberg, <a href="https://arxiv.org/abs/1802.09805">Atoms for signed permutations</a>, arXiv:1802.09805 [math.CO], 2018.
%H Alejandro H. Morales, I. Pak, and G. Panova, <a href="https://web.archive.org/web/20171109075224/http://math.ucla.edu/~ahmorales/papers/EulerFib4.pdf">Why is pi < 2 phi?</a>, Preprint, 2016; The American Mathematical Monthly, Volume 125, 2018 - Issue 8.
%H Alan H. Rapoport (proposer), <a href="http://www.jstor.org/stable/2687550">Solution to Problem 639: A Square Young Tableau</a>, College Mathematics Journal, Vol. 30 (1999), no. 5, pp. 410-411.
%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F a(n) = (n^2)! / (Product_{k=1..2n-1} k^(n - |n-k|).
%F a(n) = 0!*1!*...*(k-1)! *(k*n)! / ( n!*(n+1)!*...*(n+k-1)! ) for k=n.
%F a(n) = A088020(n)/A107254(n) = A088020(n)*A000984(n)/A079478(n). - _Henry Bottomley_, May 14 2005
%F a(n) = A153452(prime(n)^n). - _Naohiro Nomoto_, Jan 01 2009
%F 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
%F From _Peter Luschny_, May 20 2019: (Start)
%F 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.
%F a(n) = A127223(n) / A107252(n). (End)
%F 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
%F a(n+2) = (n+2) * A060856(n+1) for n >= 0. - _Tom Copeland_, May 30 2022
%e Using the hook length formula, a(4) = (16)!/(7*6^2*5^3*4^4*3^3*2^2) = 24024.
%p a:= n-> (n^2)! *mul(k!/(n+k)!, k=0..n-1):
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Apr 10 2012
%t 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 *)
%o (PARI) a(n)=(n^2)!*prod(k=0,n-1,k!/(n+k)!)
%o (Magma)
%o A039622:= func< n | n eq 0 select 1 else Factorial(n^2)*(&*[Factorial(j)/Factorial(n+j): j in [0..n-1]]) >;
%o [A039622(n): n in [0..12]]; // _G. C. Greubel_, Apr 21 2021
%o (Sage)
%o def A039622(n): return factorial(n^2)*product( factorial(j)/factorial(n+j) for j in (0..n-1))
%o [A039622(n) for n in (0..12)] # _G. C. Greubel_, Apr 21 2021
%Y Main diagonal of A060854.
%Y Also a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).
%Y Cf. A000178, A000984, A079478, A088020, A107252, A107254, A127223, A153452.
%K nonn,nice,easy
%O 0,3
%A _Floor van Lamoen_
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