A079402 Number of permutations of n^2 distinct integers free of any monotonic increasing or decreasing (n+1)-subsequence.
1, 1, 4, 1764, 577152576, 491609948246960400, 2794390432234620616607526201600, 225695005480541203944756162668572542540719673600, 487587121568323060029971679617336086880787102621519060769151477760000
Offset: 0
Examples
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}.
References
- Martin Gardner, Riddles of The Sphinx, MAA, NML vol. 32, 1987, p. 6.
- 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]
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..20
- Stanley Rabinowitz (proposer), Richard Stanley (solver), Advanced Problem 5641, Amer. Math. Monthly 76 (1969), no. 10, 1153.
- Vincent Vatter, An Erdős--Hajnal analogue for permutation classes, arXiv:1511.01076 [math.CO], 2015.
- Eric Weisstein's World of Mathematics, Erdos-Szekeres Theorem
- Wikipedia, Robinson-Schensted correspondence
- Index entries for sequences related to Young tableaux.
Programs
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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
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Maple
a:= n-> ((n^2)! *mul(k!/(n+k)!, k=0..n-1))^2: seq(a(n), n=0..8); # Alois P. Heinz, Jan 25 2014
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Mathematica
Table[( (n^2)! Product[k!/(n+k)!, {k,0,n-1}] )^2, {n,0,5}] (* Wouter Meeussen, Jan 25 2014 *) Table[((n^2)!*BarnesG[n+1]^2/BarnesG[2n+1])^2, {n, 0, 10}] (* G. C. Greubel, May 04 2021 *) -
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
Formula
a(n) = ((n^2)! * Product_{k=0..n-1} k!/(n+k)!)^2.
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
a(n) = ( (n^2)!*BarnesG(n+1)^2/BarnesG(2*n+1) )^2, where BarnesG(n) = A000178(n). - G. C. Greubel, May 04 2021
Extensions
Better name from Wouter Meeussen, Jan 25 2014
Comments