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A217325 Number of self-inverse permutations in S_n with longest increasing subsequence of length 5.

%I A217325 #19 Mar 27 2025 11:07:20
%S A217325 1,5,29,127,583,2446,10484,43363,181546,748840,3114308,12878441,
%T A217325 53594473,222761422,930856456,3893811380,16365678160,68937445765,
%U A217325 291656714515,1237403762663,5271285939671,22524961082326,96620152734652,415768621923904,1795530067804295
%N A217325 Number of self-inverse permutations in S_n with longest increasing subsequence of length 5.
%C A217325 Also the number of Young tableaux with n cells and 5 rows.
%H A217325 Alois P. Heinz, <a href="/A217325/b217325.txt">Table of n, a(n) for n = 5..1449</a> (terms 501..1000 from Seiichi Manyama)
%F A217325 a(n) = A182172(n,5) - A182172(n,4) = A049401(n) - A005817(n).
%e A217325 a(5) = 1: 12345.
%e A217325 a(6) = 5: 123465, 123546, 124356, 132456, 213456.
%p A217325 a:= proc(n) option remember; `if`(n<5, 0, `if`(n=5, 1,
%p A217325      ((n+3)*(166075637*n^5+3319452867*n^4+10706068615*n^3-39910302747*n^2
%p A217325        -182846631872*n-159926209260)*a(n-1) +(840221898216*n+133982123900
%p A217325        -322021480097*n^3-83890810854*n^4+12016871251*n^5+3735622433*n^6
%p A217325        +111397917411*n^2)*a(n-2)-(n-2)*(2142183361*n^5+66617759078*n^4
%p A217325        -47640468971*n^3-611402096064*n^2+15449945364*n+452645243780)*a(n-3)
%p A217325        -(n-2)*(n-3)*(33769818805*n^4-54918997862*n^3 -469629276839*n^2
%p A217325        +789889969148*n +94438295920)*a(-4+n) -4*(n-2)*(n-3)*(-4+n)*
%p A217325        (2060107324*n^3 -87569131518*n^2+293565842963*n -151080184425)*a(n-5)
%p A217325        +240*(n-2)*(n-3)*(n-5)*(168175627*n-312397451)*(-4+n)^2*a(n-6))/
%p A217325        (8*(13927136*n+37088781)*(n-5)*(n+6)*(n+4)*(n+3)^2)))
%p A217325     end:
%p A217325 seq(a(n), n=5..40);
%Y A217325 Column k=5 of A047884.
%Y A217325 Cf. A005817, A049401, A182172.
%K A217325 nonn
%O A217325 5,2
%A A217325 _Alois P. Heinz_, Sep 30 2012