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%I #43 May 04 2021 19:02:22
%S 1,1,4,1764,577152576,491609948246960400,
%T 2794390432234620616607526201600,
%U 225695005480541203944756162668572542540719673600,487587121568323060029971679617336086880787102621519060769151477760000
%N Number of permutations of n^2 distinct integers free of any monotonic increasing or decreasing (n+1)-subsequence.
%C Conjecture: this is equal to the number of permutations of n^2 distinct integers free of any monotonic increasing or decreasing (n+1)-subsequence. (By the Erdos-Szekeres Theorem, every permutation of n^2+1 distinct integers has such a subsequence.) - _Joseph Myers_, Jan 04 2003
%C Claude Lenormand (claude.lenormand(AT)free.fr) confirms this conjecture. - Jan 06, 2002.
%C a(n) is equal to the number of permutations of n^2 distinct integers having no monotonic sequences of length more than n. - Michael Lugo (mlugo(AT)math.upenn.edu), Mar 25 2009
%C By Robinson-Schensted correspondance, equals the square of the number of square standard Young tableaux. - _Wouter Meeussen_, Jan 25 2014
%D Martin Gardner, Riddles of The Sphinx, MAA, NML vol. 32, 1987, p. 6.
%D D. E. Knuth, The Art of Computer Programming, Vol. 3: Sorting ang Searching, Addison-Wesley, 1973, p. 69. [From Michael Lugo (mlugo(AT)math.upenn.edu), Mar 25 2009]
%H Alois P. Heinz, <a href="/A079402/b079402.txt">Table of n, a(n) for n = 0..20</a>
%H Stanley Rabinowitz (proposer), Richard Stanley (solver), <a href="http://www.jstor.org/stable/2317210">Advanced Problem 5641</a>, Amer. Math. Monthly 76 (1969), no. 10, 1153.
%H Vincent Vatter, <a href="http://arxiv.org/abs/1511.01076">An Erdős--Hajnal analogue for permutation classes</a>, arXiv:1511.01076 [math.CO], 2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erdos-SzekeresTheorem.html">Erdos-Szekeres Theorem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Robinson-Schensted_correspondence">Robinson-Schensted correspondence</a>
%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F a(n) = ((n^2)! * Product_{k=0..n-1} k!/(n+k)!)^2.
%F a(n) = (A067700(n)/2)^2 = A039622(n)^2.
%F a(n) ~ Pi * n^(2*n^2+11/6) * exp(n^2 + 1/6) / (A^2 * 2^(4*n^2-7/6)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Dec 17 2016
%F a(n) = ( (n^2)!*BarnesG(n+1)^2/BarnesG(2*n+1) )^2, where BarnesG(n) = A000178(n). - _G. C. Greubel_, May 04 2021
%e The case n=2: only a(2)=4 of the 24 permutations of {1,2,3,4} are devoid of any 3-term increasing or decreasing subsequence, namely {2,1,4,3}, {2,4,1,3}, {3,1,4,2}, {3,4,1,2}.
%p a:= n-> ((n^2)! *mul(k!/(n+k)!, k=0..n-1))^2:
%p seq(a(n), n=0..8); # _Alois P. Heinz_, Jan 25 2014
%t Table[( (n^2)! Product[k!/(n+k)!, {k,0,n-1}] )^2, {n,0,5}] (* _Wouter Meeussen_, Jan 25 2014 *)
%t Table[((n^2)!*BarnesG[n+1]^2/BarnesG[2n+1])^2, {n, 0, 10}] (* _G. C. Greubel_, May 04 2021 *)
%o (Magma) [n eq 0 select 1 else ( Round(Factorial(n^2)*(&*[ Gamma(j+1)/Gamma(n+j+1): j in [0..n-1]])) )^2: n in [0..10]]; // _G. C. Greubel_, May 04 2021
%o (Sage) [( factorial(n^2)*product( gamma(j+1)/gamma(n+j+1) for j in (0..n-1) ) )^2 for n in (0..10)] # _G. C. Greubel_, May 04 2021
%Y Cf. A000178, A039622, A067700.
%K nonn,easy
%O 0,3
%A _Dean Hickerson_, Jan 06 2003
%E Better name from _Wouter Meeussen_, Jan 25 2014
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