A138159 Triangle read by rows: T(n,k) is the number of permutations of [n] having k occurrences of the pattern 321 (n>=1, 0<=k<=n(n-1)(n-2)/6).

1, 1, 2, 5, 1, 14, 6, 3, 0, 1, 42, 27, 24, 7, 9, 6, 0, 4, 0, 0, 1, 132, 110, 133, 70, 74, 54, 37, 32, 24, 12, 16, 6, 6, 8, 0, 0, 5, 0, 0, 0, 1, 429, 429, 635, 461, 507, 395, 387, 320, 260, 232, 191, 162, 104, 130, 100, 24, 74, 62, 18, 32, 10, 30, 13, 8, 0, 10, 10, 0, 0, 0, 6, 0, 0, 0, 0, 1

Offset: 0

Author

Emeric Deutsch, Mar 27 2008

Keywords

• nonn
• look
• tabf

Comments

Row n has 1 + n(n-1)(n-2)/6 terms.

Sum of row n is n! (A000142).

T(n,0) = A000108(n) (the Catalan numbers).

T(n,1) = A003517(n-1).

T(n,2) = A001089(n).

Sum_{k>=0} k * T(n,k) = A001810(n).

Examples

			T(4,2) = 3 because we have 4312, 4231 and 3421.
Triangle starts:
    1;
    1;
    2;
    5,   1;
   14,   6,   3,  0,  1;
   42,  27,  24,  7,  9,  6,  0,  4,  0,  0,  1;
  132, 110, 133, 70, 74, 54, 37, 32, 24, 12, 16, 6, 6, 8, 0, 0, 5, 0, 0, 0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..15, flattened
• D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
• FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3
• M. Fulmek, Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three, Adv. Appl. Math., 30, 2003, 607-632. also Arxiv CO/0112092.
• Toufik Mansour, Sherry H. F. Yan and Laura L. M. Yang, Counting occurrences of 231 in an involution, Discrete Mathematics 306 (2006), pages 564-572.
• J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math. 152 (1996), no. 1-3, 307-313.
• J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
• J. Noonan and D. Zeilberger, The enumeration of permutations with a prescribed number of "forbidden" patterns, Adv. Appl. Math., 17, 1996, 381-407.

Crossrefs

Cf. A000108, A000142, A000292, A003517, A001089, A001810, A056986, A263771.

Programs

Maple

# The following Maple program yields row 9 of the triangle; change the value of n to obtain other rows.
n:=9: with(combinat): P:=permute(n): f:=proc(k) local L: L:=proc(j) local ct, i: ct:=0: for i to j-1 do if P[k][j] < P[k][i] then ct:=ct+1 else end if end do: ct end proc: add(L(j)*(L(j)+P[k][j]-j),j=1..n) end proc: a:=sort([seq(f(k),k=1..factorial(n))]): for h from 0 to (1/6)*n*(n-1)*(n-2) do c[h]:=0: for m to factorial(n) do if a[m]=h then c[h]:=c[h]+1 else end if end do end do: seq(c[h],h=0..(1/6)*n*(n-1)*(n-2));
# second Maple program:
b:= proc(s, c) option remember; (n-> `if`(n=0, x^c, add(b(s minus {j},
      (t-> (j-n+t)*t+c)(nops(select(x-> x>j, s)))), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..9);  # Alois P. Heinz, Dec 01 2021

Mathematica

ro[n_] := With[{}, P = Permutations[Range[n]]; f[k_] := With[{}, L[j_] := With[{}, ct = 0; Do[If[P[[k, j]] < P[[k, i]], ct = ct + 1], {i, 1, j - 1}]; ct]; Sum[L[j]*(L[j] + P[[k, j]] - j), {j, 1, n}]]; a = Sort[Table[f[k], {k, 1, n!}]]; Do[c[h] = 0; Do[If[a[[m]] == h, c[h] = c[h] + 1], {m, 1, n!}], {h, 0, (1/6)*n*(n - 1)*(n - 2)}]; Table[c[h], {h, 0, (1/6)*n*(n - 1)*(n - 2)}]]; Flatten[Table[ro[n], {n, 1, 7}]] (* Jean-François Alcover, Sep 01 2011, after Maple *)

Formula

The number of 321-patterns of a given permutation p of [n] is given by Sum(L[i]R[i],i=1..n), where L (R) is the left (right) inversion vector of p. L and R are related by R[i]+i=p[i]+L[i] (the given Maple program makes use of this approach). References contain formulas and generating functions for the first few columns (some are only conjectured).