A082970
Number of permutations of length n containing 2 occurrences of 132.
Original entry on oeis.org
4, 23, 107, 464, 1950, 8063, 33033, 134576, 546312, 2212550, 8946454, 36134656, 145831270, 588199815, 2371435125, 9557736480, 38511326040, 155143873170, 624899673690, 2516678580000, 10134353299980, 40805797511622
Offset: 4
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Table[Binomial[2n-6,n-2] (n^3+17n^2-80n+80)/(2n(n-1)),{n,4,30}] (* Harvey P. Dale, Dec 25 2018 *)
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a(n)=binomial(2*n-6,n-2)*(n^3+17*n^2-80*n+80)/2/n/(n-1)
A082971
Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.
Original entry on oeis.org
1, 14, 82, 410, 1918, 8657, 38225, 166322, 716170, 3059864, 12994936, 54924212, 231235054, 970347575, 4060697955, 16952812170, 70629116910, 293720506860, 1219498444500, 5055891511980, 20933654593020, 86571545598642, 357628915621698, 1475896409177780
Offset: 4
a(4)=1 because we have 1432 (the 132 occurrences are 143, 142 and 132).
- Vincenzo Librandi, Table of n, a(n) for n = 4..1000
- Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
- Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
- T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.
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[1] cat [(n^6+51*n^5-407*n^4-99*n^3+7750*n^2-22416*n+20160)* Factorial(2*n-9)/(6*Factorial(n)*Factorial(n-5)): n in [5..30]]; // Vincenzo Librandi, Oct 30 2018
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P:=2*x^3-5*x^2+7*x-2: Q:=-22*x^6-106*x^5+292*x^4-302*x^3+135*x^2-27*x+2: g:= (P+Q/(1-4*x)^(5/2))*1/2: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=4..25); # Emeric Deutsch, Mar 27 2008
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a[4] = 1; a[n_] := (n^6 + 51 n^5 - 407 n^4 - 99 n^3 + 7750 n^2 - 22416 n + 20160) (2 n - 9)!/(6 n! (n - 5)!);
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Oct 30 2018 *)
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a(n)=(2*n-9)!/n!/6/(n-5)!*(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n+20160)
A138162
Number of permutations of {1,2,...,n} containing exactly 4 occurrences of the 132 pattern.
Original entry on oeis.org
12, 96, 526, 2593, 12165, 55482, 248509, 1099255, 4817998, 20968680, 90747564, 390927869, 1677551078, 7174848666, 30598014925, 130155932685, 552386655300, 2339526458640, 9890067346740, 41737405295250, 175859194700958
Offset: 5
a(5)=12 because we have 12534, 12453, 14253, 14523, 13254, 13524, 15324, 14352, 31542, 21534, 21453 and 25143.
- Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
- Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
- T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.
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P:=5*x^4-7*x^3+2*x^2+8*x-3: Q:=2*x^9+218*x^8+1074*x^7-1754*x^6 +388*x^5 +1087*x^4-945*x^3+320*x^2-50*x+3: g:=(P+Q/(1-4*x)^(7/2))*1/2: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=5..25);
A138163
Number of permutations of {1,2,...,n} containing exactly 5 occurrences of the pattern 132.
Original entry on oeis.org
5, 55, 394, 2225, 11539, 57064, 273612, 1283621, 5924924, 27005978, 121861262, 545368160, 2423923480, 10710273856, 47085144255, 206085075295, 898489543020, 3903621095130, 16906888008960, 73018012573950, 314540265217362
Offset: 5
a(5)=5 because we have 13542, 14532, 15243, 15342 and 15423.
- B. K. Nakamura, Computational methods in permutation patterns, PhD Dissertation, Rutgers University, May 2013.
- Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.
- Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
- T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.
- B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, PU. M. A. Vol. 24 2013 No 2, pp. 179-194.
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a:=proc(n) options operator, arrow: (1/120)*(n^12+170*n^11 +1861*n^10 -88090*n^9 -307617*n^8 +27882510*n^7 -348117457*n^6 +2119611370*n^5 -6970280884*n^4 +10530947320*n^3 +2614396896*n^2 -30327454080*n +29059430400) *factorial(2*n-15) / (factorial(n)*factorial(n-7)) end proc: 5, 55, 394, seq(a(n), n = 8 .. 25);
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terms = 21; offset = 5;
P[x_] := 14 x^5 - 17 x^4 + x^3 - 16 x^2 + 14 x - 2;
Q[x_] := -50 x^11 - 2568 x^10 - 10826 x^9 + 16252 x^8 - 12466 x^7 + 16184 x^6 - 16480 x^5 + 9191 x^4 - 2893 x^3 + 520 x^2 - 50 x + 2;
Drop[CoefficientList[(1/2) (P[x] + Q[x]/(1 - 4 x)^(9/2)) + O[x]^(terms + offset), x], offset] (* Jean-François Alcover, Dec 13 2017 *)