login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A052399 Number of permutations in S_n with longest increasing subsequence of length <= 6. 9

%I

%S 1,1,2,6,24,120,720,5039,40270,361302,3587916,38957991,457647966,

%T 5763075506,77182248916,1091842643475,16219884281650,251774983140578,

%U 4066273930979460,68077194367392864,1177729684507324152,20995515989327134152,384762410996641402384

%N Number of permutations in S_n with longest increasing subsequence of length <= 6.

%C Previous name was: Related to Young tableaux of bounded height.

%H Alois P. Heinz, <a href="/A052399/b052399.txt">Table of n, a(n) for n = 0..250</a>

%H F. Bergeron and F. Gascon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/CYT/cyt.html">Counting Young tableaux of bounded height</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.7.

%H Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.

%H Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, <a href="http://arxiv.org/abs/1504.02513">The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r</a>, arXiv:1504.02513 [math.CO], 2015.

%H Nathaniel Shar, <a href="https://pdfs.semanticscholar.org/98e3/71b675789ed6ec4f9c9cd82e2dee9ca79399.pdf">Experimental methods in permutation patterns and bijective proof</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>

%F a(n) ~ 5 * 2^(2*n + 6) * 3^(2*n + 21) / (n^(35/2) * Pi^(5/2)). - _Vaclav Kotesovec_, Sep 10 2014

%p h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j

%p +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)

%p end:

%p g:= proc(n, i, l) option remember;

%p `if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1$n])^2,

%p g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))

%p end:

%p a:= n-> g(n, 6, []):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Apr 10 2012

%p # second Maple program

%p a:= proc(n) option remember; `if`(n<7, n!,

%p ((56*n^5-9408+11032*n+19028*n^2+7360*n^3+1092*n^4)*a(n-1)

%p -4*(196*n^3+1608*n^2+3167*n+444)*(n-1)^2*a(n-2)

%p +1152*(2*n+3)*(n-1)^2*(n-2)^2*a(n-3))/ ((n+9)*(n+8)^2*(n+5)^2))

%p end:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 26 2012

%t h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 6], {n, 0, 30}] (* _Jean-François Alcover_, Mar 11 2014, after _Alois P. Heinz_ *)

%Y Cf. A005802, A047889, A047890.

%Y Column k=6 of A214015.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Mar 13 2000

%E More terms from _Alois P. Heinz_, Apr 10 2012

%E New name from _Vaclav Kotesovec_, Sep 10 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 16:05 EST 2022. Contains 358701 sequences. (Running on oeis4.)