login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 15
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 29 23:12 EST 2020. Contains 338779 sequences. (Running on oeis4.)