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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
Eric Weisstein's World of Mathematics, Permutation Pattern
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.
Sequence in context: A159924 A133367 A179456 * A121575 A121576 A049444
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 05 2012
STATUS
approved

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Last modified May 25 15:42 EDT 2024. Contains 372799 sequences. (Running on oeis4.)