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A129777 Number of freely-braided hexagon-avoiding permutations in S_n; the freely-braided hexagon-avoiding permutations are those that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.

%I A129777 #19 Jan 12 2025 17:12:54
%S A129777 1,1,2,6,20,71,260,971,3670,13968,53369,204352,783408,3005284,
%T A129777 11533014,44267854,169935041,652385639,2504613713,9615798516,
%U A129777 36917689075,141737959416,544175811783,2089262741393,8021347093432,30796530585417,118237818141689,453953210838465
%N A129777 Number of freely-braided hexagon-avoiding permutations in S_n; the freely-braided hexagon-avoiding permutations are those that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
%C A129777 If w is freely-braided and hexagon-avoiding, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w}.
%D A129777 Jozsef Losonczy, Maximally clustered elements and Schubert varieties,  Ann. Comb. 11 (2007), no. 2, 195-212.
%H A129777 H. Denoncourt and B. Jones, <a href="http://arXiv.org/abs/0704.3469">The enumeration of maximally clustered permutations</a>.
%H A129777 B. Jones, <a href="http://www.arXiv.org/abs/0704.3067">Kazhdan--Lusztig polynomials for maximally-clustered hexagon-avoiding permutations</a>.
%H A129777 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6, -9, 3, 1, -8, -1, 1).
%F A129777 G.f.: 1+(-x^7-2x^6+2x^5+x^4-3x^3+4x^2-x) / (x^7-x^6-8x^5+x^4+3x^3-9x^2+6x-1).
%e A129777 a(8)=3670 because there are 3670 permutations of size 8 that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
%t A129777 LinearRecurrence[{6, -9, 3, 1, -8, -1, 1}, {1, 2, 6, 20, 71, 260, 971}, 27] (* _Jean-François Alcover_, Feb 02 2019 *)
%o A129777 (PARI) lista(nt) = { my(x = 'x + 'x*O('x^nt) ); P = (-x^7-2*x^6+2*x^5+x^4-3*x^3+4*x^2-x) / (x^7-x^6-8*x^5+x^4+3*x^3-9*x^2+6*x-1); print(Vec(P));}  \\ _Michel Marcus_, Mar 17 2013
%Y A129777 Cf. A058094, A108600.
%K A129777 nonn
%O A129777 0,3
%A A129777 Brant Jones (brant(AT)math.washington.edu), May 17 2007
%E A129777 More terms from _Michel Marcus_, Mar 17 2013
%E A129777 a(0)=1 prepended by _Alois P. Heinz_, Jan 12 2025