A138163 -id:A138163 - cp's OEIS frontend

A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

1, 1, 2, 5, 1, 14, 5, 4, 1, 42, 21, 23, 14, 12, 5, 3, 132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2, 429, 330, 464, 410, 526, 394, 475, 365, 360, 298, 281, 175, 206, 126, 93, 55, 23, 14, 13, 1, 2, 1430, 1287, 1950, 1918, 2593, 2225, 2858, 2489, 2682, 2401

Offset: 0

Christian Stump, Oct 26 2015

Row sums give A000142.
First column gives A000108.
Also the number of permutations of n and k occurrences of either of the fixed pattern 132, 213, 231 (these are all connected by reverses and inverses).
Columns k=1-5 give: A002054(n-2) for n>=3, A082970, A082971, A138162, A138163. - Alois P. Heinz, Oct 27 2015

			Triangle begins:
    1;
    1;
    2;
    5,  1;
   14,  5,   4,  1;
   42, 21,  23, 14, 12,  5,  3;
  132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2;
  ...

Alois P. Heinz, Rows n = 0..10, flattened
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.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern 312 in a permutation, The number of occurrences of the pattern 213 in a permutation, The number of occurrences of the pattern 231 in a permutation, The number of occurrences of the pattern 132 in a permutation
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Cf. A000108, A000142, A001810, A002054, A082970, A082971, A138162, A138159, A138163.

Join@@Array[Table[Length@Select[Permutations@Range@#,Length@Select[Subsets[#,{3}],Ordering@Ordering@#=={3,1,2}&]==k&],{k,0,Binomial[#+1,3]}]//.{a__,0}:>{a}&,8,0]  (* Giorgos Kalogeropoulos, Mar 26 2021 *)

Sum_{k>0} k * T(n,k) = A001810(n). - Alois P. Heinz, Oct 27 2015

More terms from Alois P. Heinz, Oct 26 2015

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

Benoit Cloitre, May 27 2003

			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.

Cf. A002054, A082970, A138162, A138163.

Column k=3 of A263771.

[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

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

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 *)

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)

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).

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)!) for n>=5; a(4)=1. G.f.: (1/2)*(P(x) + Q(x)/(1-4*x)^(5/2)), where P(x) = 2*x^3 - 5*x^2 + 7*x - 2, Q(x) = -22*x^6 - 106*x^5 + 292*x^4 - 302*x^3 + 135*x^2 - 27*x + 2. - Emeric Deutsch, Mar 27 2008

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

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

Emeric Deutsch, Mar 27 2008

nonn

			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.

Cf. A002054, A082970, A082971, A138163.

Column k=4 of A263771.

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);

a(n) = (n^9+102n^8-282n^7-12264n^6+32589n^5+891978n^4-7589428n^3 +25452024n^2-39821760n +23950080)(2n-12)!/[24n!(n-6)! ] for n>=6, a(5)=12.

G.f.: (1/2)[P(x) + Q(x)/(1-4x)^(7/2)], where P(x)=5x^4-7x^3+2x^2+8x-3, Q(x)=2x^9 +218x^8+1074x^7 -1754x^6 +388x^5 +1087x^4 -945x^3+320x^2-50x+3.

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A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

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A082971 Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.

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A138162 Number of permutations of {1,2,...,n} containing exactly 4 occurrences of the 132 pattern.

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