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A180389 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) the same as number of rises in the inverse permutation. 3

%I #41 Jun 11 2015 15:11:53

%S 1,1,2,6,22,96,492,2952,20588,164990,1497740,15187692,169974040,

%T 2078905752,27567259896,393759207372,6025346314756,98317949671110,

%U 1703879074519500,31251488731748108,604748393942784976,12312387380060084768,263079571362773145632

%N Number of permutations of 1..n with number of rises (p(i+1)>p(i)) the same as number of rises in the inverse permutation.

%C Also equals sum of squares of the coefficients of the (numerators of) the G.F. for the count of monomials in the Schur polynomials of degree n (all partitions of weight n), in function of the number of variables v. - _Wouter Meeussen_, Dec 27 2010

%C Studied by Carlitz, Roselle, and Scoville in 'Permutations and Sequences with Repetitions by Number of Increases'. They refer to a rise/ascent as a 'jump', and consider the first entry of a permutation to always be a jump, so #jumps=#rises+1. Similarly, the number of rises in the inverse permutation/number of inverse descents corresponds to what they call the 'number of readings', and follow a convention so that #rises in inverse permutation+1=#readings. Formula can be attained by R_m(k,r), setting t=k, and summing k from 1 to m+1. - _Kevin Dilks_, Jun 09 2015

%H Vaclav Kotesovec, <a href="/A180389/b180389.txt">Table of n, a(n) for n = 0..200</a>

%H L. Carlitz et al., <a href="http://dx.doi.org/10.1016/S0021-9800(66)80057-1">Permutations and sequences with repetitions by number of increases</a>, J. Combin. Theory, 1 (1966), 350-374.

%H Wouter Meeussen, <a href="http://users.telenet.be/Wouter.Meeussen/SchurPolys3.xls">Schur Polynomials</a>

%H Wouter Meeussen, <a href="/A180389/a180389_1.txt">Some notes on the link between A180389 and Schur polynomials</a>

%F a(n) = Sum_{m=0..n+1} Sum_{i=0..m} Sum_{j=0..m} (-1)^{i+j} binomial(n+1,i) binomial(n+1,j) binomial((m-i)*(m-j)+n-1,n). - _Kevin Dilks_, Jun 09 2015

%F a(n) ~ sqrt(3) * n! / sqrt(Pi*n). - _Vaclav Kotesovec_, Jun 10 2015

%e For n=4, a(4)=22 are all permutations of length 4 except for 3142 (which has only one ascent, and two inverse ascents) and 2413 (which has two ascents, and only one inverse ascent). - _Kevin Dilks_, Jun 09 2015

%p seq(add(add(add((-1)^(i+j)*binomial(n+1,i)*binomial(n+1,j)*binomial((m-i)*(m-j)+n-1,n),i=0..m),j=0..m),m=0..n+1), n=0..30); # _Kevin Dilks_, Jun 09 2015

%t Table[Sum[Sum[Sum[(-1)^(i+j)*Binomial[n+1,i]*Binomial[n+1,j]*Binomial[(m-i)*(m-j)+n-1,n],{i,0,m}],{j,0,m}],{m,0,n+1}],{n,0,10}] (* _Kevin Dilks_, Jun 09 2015 *)

%Y A180388(n) + a(n) = n! = A000142(n).

%K nonn

%O 0,3

%A _Leroy Quet_, _D. S. McNeil_ and _R. H. Hardin_ in the Sequence Fans Mailing List Sep 01 2010

%E a(15)-a(20) from _Wouter Meeussen_, Dec 27 2010

%E a(0)=1 prepended by _Alois P. Heinz_, Jun 10 2015

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)