A214152 Number of permutations T(n,k) in S_n containing an increasing subsequence of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 2, 1, 6, 5, 1, 24, 23, 10, 1, 120, 119, 78, 17, 1, 720, 719, 588, 207, 26, 1, 5040, 5039, 4611, 2279, 458, 37, 1, 40320, 40319, 38890, 24553, 6996, 891, 50, 1, 362880, 362879, 358018, 268521, 101072, 18043, 1578, 65, 1, 3628800, 3628799, 3612004, 3042210, 1438112, 337210, 40884, 2603, 82, 1

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..55, flattened

Eric Weisstein's World of Mathematics, Permutation Pattern

Wikipedia, Longest increasing subsequence problem

Wikipedia, Young tableau

Formula

T(n,k) = Sum_{i=k..n} A047874(n,i).

T(n,k) = A000142(n) - A214015(n,k-1).

Example

T(3,2) = 5.  All 3! = 6 permutations of {1,2,3} contain an increasing subsequence of length 2 with the exception of 321.
Triangle T(n,k) begins:
:    1;
:    2,    1;
:    6,    5,    1;
:   24,   23,   10,    1;
:  120,  119,   78,   17,   1;
:  720,  719,  588,  207,  26,  1;
: 5040, 5039, 4611, 2279, 458, 37,  1;

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= (n, k)-> n! -g(n, k-1, []):
seq(seq(T(n, k), k=1..n), n=1..12);

Mathematica

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] ]; g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; t[n_, k_] := n! - g[n, k-1, {}]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

Crossrefs

Columns k=1-10 give: A000142 (for n>0), A033312, A056986, A158005, A158432, A159139, A159175, A217675, A217676, A217677.

Row sums give: A003316.

T(2n,n) gives A269021.

Diagonal and lower diagonals give: A000012, A002522, A217200, A217193.

Cf. A047874, A214015.

Sequence in context: A159924 A133367 A179456 * A121575 A121576 A049444

Adjacent sequences: A214149 A214150 A214151 * A214153 A214154 A214155

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 05 2012

Status

approved