A094112 Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding the 123-, the 132- and the 231-pattern is equal to k.

1, 0, 2, 0, 3, 3, 0, 12, 8, 4, 0, 60, 40, 15, 5, 0, 360, 240, 90, 24, 6, 0, 2520, 1680, 630, 168, 35, 7, 0, 20160, 13440, 5040, 1344, 280, 48, 8, 0, 181440, 120960, 45360, 12096, 2520, 432, 63, 9, 0, 1814400, 1209600, 453600, 120960, 25200, 4320, 630, 80, 10, 0

Offset

1,3

Comments

Row sums are the factorial numbers (A000142).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)

Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.

Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.

E. Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.

R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985.

Formula

T(n, k) = n!/[(k-2)!k] for 2<=k<=n-1; T(n, n)=n; T(n, 1)=0 for n>=2; T(n, k)=0 for k>n.

G.f.: sum(T(n, k)t^k z^n/n!, n, k>=1) = z[(t-1)exp(tz)+1]/(1-z).

Example

T(4,3)=8 because the permutations 2134, 2143, 3124, 3142, 3241, 4123, 4132 and 4231 do not avoid all three patterns 123, 132 and 231, but their initial segments of length three, namely 213, 214, 312, 314, 324, 412, 413 and 423, do.
Triangle begins:
1;
0,2;
0,3,3;
0,12,8,4;
0,60,40,15,5;
0,360,240,90,24,6;
...

Maple

T:=proc(n, k) if n=1 and k=1 then 1 elif n=1 then 0 elif k=1 then 0 elif k=n then n elif k>1 and k<n then n!/(k-2)!/k else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..11);

Mathematica

T[n_, k_] := Which[n == 1 && k == 1, 1, n == 1, 0, k == 1, 0, k == n, n, k > 1 && k < n, n!/(k-2)!/k, True, 0]; Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 22 2019, from PARI *)

Crossrefs

Cf. A000142.

Sequence in context: A298605 A180013 A094067 * A333303 A326926 A347564

Adjacent sequences: A094109 A094110 A094111 * A094113 A094114 A094115

Keyword

nonn,tabl

Author

Emeric Deutsch, May 31 2004

Status

approved