A003316
Sum of lengths of longest increasing subsequences of all permutations of n elements.
Original entry on oeis.org
1, 3, 12, 58, 335, 2261, 17465, 152020, 1473057, 15730705, 183571817, 2324298010, 31737207034, 464904410985, 7272666016725, 121007866402968, 2133917906948645, 39756493513248129, 780313261631908137, 16093326774432620874, 347958942706716524974
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alois P. Heinz, Table of n, a(n) for n = 1..80
- R. M. Baer and P. Brock, Natural sorting over permutation spaces, Math. Comp. 22 1968 385-410.
- R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
- A. M. Vershik and S. V. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux, Doklady Akademii Nauk SSSR, 1977, Volume 233, Number 6, Pages 1024-1027. In Russian.
- Wikipedia, Longest increasing subsequence
Cf.
A008304 (which is concerned with runs of adjacent elements).
-
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))):
a:= n-> add(k* (g(n-k, k, [k])), k=1..n):
seq(a(n), n=1..22); # Alois P. Heinz, Jul 05 2012
-
h[l_List] := Module[{n = Length[l]}, Total[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}]]]; a[n_] := Sum[k*g[n-k, k, {k}], {k, 1, n}]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *)
Corrected a(13) and extended beyond a(16) by
Alois P. Heinz, Jul 05 2012
A321273
Sum over all permutations of [n] of the maximum of the lengths of increasing or decreasing subsequences.
Original entry on oeis.org
1, 4, 14, 70, 396, 2628, 20270, 175392, 1686374, 17920528, 208454628, 2629931688, 35774761662, 522351495684, 8149929922408, 135284126840592, 2380119357533974, 44243729657494640, 866599471539160876, 17839886344238238784, 385065445154671172880, 8695565142604747421416
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*max(l[1], nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
-
h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]][Length[l]];
f[l_] := h[l]^2 Max[l[[1]], Length[l]];
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz *)
A321277
One half of the sum over all permutations of [n] of the absolute difference between the length of the longest increasing subsequence and the length of the longest decreasing subsequence.
Original entry on oeis.org
0, 1, 2, 12, 61, 367, 2805, 23372, 213317, 2189823, 24882811, 305633678, 4037554628, 57447084699, 877263905683, 14276260437624, 246201450585329, 4487236144246511, 86286209907252739, 1746559569805617910, 37106502447954647906, 825196425771658993531
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*abs(l[1]-nops(l))/2:
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
A321278
One half of the sum over all permutations of [n] of the squared difference between the length of the longest increasing subsequence and the length of the longest decreasing subsequence.
Original entry on oeis.org
0, 1, 4, 18, 105, 699, 5285, 45128, 431223, 4540775, 52268029, 653096124, 8810538490, 127622293057, 1975379879871, 32537074533872, 568268861724191, 10490690233451583, 204118868130889733, 4174977363687339452, 89554055679215605982, 2010207472655266461533
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*(l[1]-nops(l))^2/2:
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
A321275
Sum over all permutations of [n] of the product of the lengths of longest increasing subsequence and longest decreasing subsequence.
Original entry on oeis.org
1, 4, 22, 132, 890, 6812, 58422, 555900, 5819658, 66554180, 825839718, 11054124886, 158795559000, 2437248222710, 39809464449676, 689538524084168, 12625142440334342, 243656361772961292, 4943801229819987022, 105212500452414418118, 2343513475564027153128
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*l[1]*nops(l):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
A321276
Sum over all permutations of [n] of the length of the longest increasing subsequence raised to the power of the length of the longest decreasing subsequence.
Original entry on oeis.org
1, 3, 20, 174, 1915, 25861, 407691, 7330188, 148016449, 3312032213, 81207824255, 2162810487154, 62125097028962, 1913156511113517, 62839800627095263, 2191735865280260976, 80859575674731497805, 3144804693463679033629, 128550453029684197431607
Offset: 1
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> h(l)^2*l[1]^nops(l):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
Showing 1-6 of 6 results.