| Displaying 1-8 of 8 results found.
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0, 0, 0, 1, 5, 20, 75, 271, 957, 3337, 11559, 39896, 137423, 472808, 1625632, 5587228, 19198971, 65963978, 226623902, 778551761, 2674604282, 9188106871, 31563807424, 108430368827, 372487292867, 1279591674070, 4395730089428
COMMENTS
A recurrence relation follows in a straightforward manner from the above formula and the recurrence relation for A058094.
FORMULA
G.f.: x^4*(1 - x + x^2 + x^3) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6). - R. J. Mathar, Dec 02 2007 a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7. - Colin Barker, Aug 21 2019
MAPLE
b[1]:=1:b[2]:=2:b[3]:=5:b[4]:=14:b[5]:=42:b[6]:=132: for n from 6 to 32 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od:a[1]:=0:a[2]:=0:a[3]:=0:for n from 4 to 32 do a[n]:=b[n]-3*b[n-1]+b[n-2] od: seq(a[n], n=1..32); # Emeric Deutsch, Apr 12 2005
PROG
(PARI) concat([0, 0, 0], Vec(x^4*(1 - x + x^2 + x^3) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019
0, 0, 0, 0, 1, 6, 25, 93, 333, 1172, 4083, 14137, 48778, 167981, 577874, 1986747, 6828120, 23462470, 80611581, 276944893, 951422603, 3268470411, 11228209786, 38572124196, 132505812826, 455192771711, 1563706508759, 5371738013650
COMMENTS
A recurrence relation follows in a straightforward manner from the above formula and the recurrence relation for A058094.
FORMULA
G.f.: x^5 / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>6.
(End)
MAPLE
b[1]:=1:b[2]:=2:b[3]:=5:b[4]:=14:b[5]:=42:b[6]:=132: for n from 6 to 34 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: a[1]:=0:a[2]:=0:a[3]:=0:a[4]:=0: for n from 5 to 34 do a[n]:=2*b[n-2]-5*b[n-3]+b[n-4]+a[n-1] od: seq(a[n], n=1..34); # Emeric Deutsch, Apr 12 2005
MATHEMATICA
LinearRecurrence[{6, -11, 9, -4, -4, 1}, {0, 0, 0, 0, 1, 6}, 40] (* Vincenzo Librandi, Aug 15 2017 *)
PROG
(PARI) concat([0, 0, 0, 0], Vec(x^5 / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019
Arises in enumeration of 321-hexagon-avoiding permutations.
+10
6
0, 0, 1, 4, 14, 48, 165, 568, 1954, 6717, 23082, 79307, 272470, 936065, 3215741, 11047122, 37950140, 130369334, 447853808, 1538496047, 5285135093, 18155807539, 62369881206, 214256590058, 736026444181, 2528439830821
FORMULA
Stankova and West give an explicit recurrence.
G.f.: x^3*(1 - 2*x + x^2 - x^3 - x^4) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7.
(End)
MAPLE
b[1]:=1: b[2]:=2: b[3]:=5: b[4]:=14: b[5]:=42: b[6]:=132: for n from 6 to 35 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: seq(b[n], n=1..35): a[1]:=0: a[2]:=0: for n from 3 to 35 do a[n]:=b[n]-2*b[n-1] od: seq(a[n], n=1..35); # here b[n]= A058094(n).
PROG
(PARI) concat([0, 0], Vec(x^3*(1 - 2*x + x^2 - x^3 - x^4) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 20 2019
Arises in enumeration of 321-hexagon-avoiding permutations.
+10
6
0, 0, 0, 0, 0, 1, 5, 19, 68, 240, 839, 2911, 10054, 34641, 119203, 409893, 1408873, 4841373, 16634350, 57149111, 196333312, 674477710, 2317047808, 7959739375, 27343914410, 93933688630, 322686958885, 1108513737048, 3808031504891
FORMULA
G.f.: x^6*(1 - x) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6).
a(n) = 6*a(n-1) - 11*a(n-2) + 9*a(n-3) - 4*a(n-4) - 4*a(n-5) + a(n-6) for n>7.
(End)
MAPLE
b[1]:=1: b[2]:=2: b[3]:=5: b[4]:=14: b[5]:=42: b[6]:=132: for n from 6 to 45 do b[n+1]:=6*b[n]-11*b[n-1]+9*b[n-2]-4*b[n-3]-4*b[n-4]+b[n-5] od: a[1]:=0: a[2]:=0: a[3]:=0: a[4]:=0: a[5]:=0: for n from 6 to 40 do a[n]:=2*b[n-3]-5*b[n-4]+b[n-5] od: seq(a[n], n=1..40); # Emeric Deutsch, Jun 08 2004
PROG
(PARI) concat([0, 0, 0, 0, 0], Vec(x^6*(1 - x) / (1 - 6*x + 11*x^2 - 9*x^3 + 4*x^4 + 4*x^5 - x^6) + O(x^30))) \\ Colin Barker, Aug 21 2019
a(n) = 4a(n-1) - 4a(n-2) + 3a(n-3) + a(n-4) - a(n-5).
+10
1
1, 2, 5, 14, 42, 128, 389, 1179, 3572, 10825, 32810, 99446, 301412, 913547, 2768863, 8392136, 25435699, 77092976, 233660832, 708201794, 2146486339, 6505777953, 19718339694, 59764246943, 181139247400, 549014312524, 1664005563066
COMMENTS
Arises in enumeration of certain pattern-avoiding permutations.
FORMULA
G.f.: x*(1 - 2*x + x^2 - x^3 - x^4)/(1 - 4*x + 4*x^2 - 3*x^3 - x^4 + x^5). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009; corrected by R. J. Mathar, Sep 16 2009]
MAPLE
a[1]:=1: a[2]:=2: a[3]:=5: a[4]:=14: a[5]:=42: for n from 6 to 32 do a[n]:=4*a[n-1]-4*a[n-2]+3*a[n-3]+a[n-4]-a[n-5] od: seq(a[j], j=1..32); # Emeric Deutsch, Apr 12 2005
MATHEMATICA
LinearRecurrence[{4, -4, 3, 1, -1}, {1, 2, 5, 14, 42}, 40] (* Harvey P. Dale, Jul 14 2024 *)
Number of maximally-clustered hexagon-avoiding permutations in S_n; the maximally-clustered hexagon-avoiding permutations are those that avoid 3421, 4312, 4321, 46718235, 46781235, 56718234, 56781234.
+10
0
1, 2, 6, 21, 78, 298, 1157, 4535, 17872, 70644, 279704, 1108462, 4395045, 17431206, 69144643, 274300461, 1088215370, 4317321235, 17128527716, 67956202025, 269612504850, 1069675361622, 4243893926396, 16837490364983, 66802139457897, 265035151393777
COMMENTS
If w is maximally-clustered and hexagon-avoiding, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w}.
REFERENCES
Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007), no. 2, 195-212.
FORMULA
G.f.: (3x^6+x^5-5x^4+7x^3-5x^2+x) / (-3x^6+4x^5+8x^4-14x^3+15x^2-7x+1).
EXAMPLE
a(8)=4535 because there are 4535 permutations of size 8 that avoid 3421, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
PROG
(PARI) lista(nt) = { my(x = 'x + 'x*O('x^nt) ); P = (3*x^6+x^5-5*x^4+7*x^3-5*x^2+x) / (-3*x^6+4*x^5+8*x^4-14*x^3+15*x^2-7*x+1); print(Vec(P)); } \\ Michel Marcus, Mar 17 2013
AUTHOR
Brant Jones (brant(AT)math.washington.edu), May 17 2007
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.
+10
0
1, 2, 6, 20, 71, 260, 971, 3670, 13968, 53369, 204352, 783408, 3005284, 11533014, 44267854, 169935041, 652385639, 2504613713, 9615798516, 36917689075, 141737959416, 544175811783, 2089262741393, 8021347093432, 30796530585417, 118237818141689, 453953210838465
COMMENTS
If w is freely-braided and hexagon-avoiding, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w}.
REFERENCES
Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007), no. 2, 195-212.
FORMULA
G.f.: (-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).
EXAMPLE
a(8)=3670 because there are 3670 permutations of size 8 that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
MATHEMATICA
LinearRecurrence[{6, -9, 3, 1, -8, -1, 1}, {1, 2, 6, 20, 71, 260, 971}, 27] (* Jean-François Alcover, Feb 02 2019 *)
PROG
(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
AUTHOR
Brant Jones (brant(AT)math.washington.edu), May 17 2007
Number of Deodhar elements in the finite Weyl group D_n.
+10
0
2, 5, 14, 48, 167, 575, 1976, 6791
COMMENTS
The Deodhar elements are a subset of the fully commutative elements. If w is Deodhar, there are simple explicit formulas for all the Kazhdan-Lusztig polynomials P_{x,w} and the Kazhdan-Lusztig basis element C'_w is the product of C'_{s_i}'s corresponding to any reduced expression for w.
REFERENCES
S. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin., 13(2):111-136, 2001.
V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata, 36(1): 95-119, 1990.
EXAMPLE
a(4)=48 because there are 48 fully commutative elements in D_4 and since the first non-Deodhar fully-commutative element does not appear until D_6, these are all of the Deodhar elements in D_4.
AUTHOR
Brant Jones (brant(AT)math.washington.edu), May 17 2007
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