A268699 Total number of sequences with c_j copies of j and longest increasing subsequence of length k summed over all compositions [c_1, c_2, ..., c_k] of n.

1, 1, 2, 6, 22, 95, 471, 2618, 16052, 107313, 775045, 6002106, 49536510, 433485429, 4004680967, 38912323570, 396393445096, 4221367056961, 46878865762185, 541660919690866, 6498811587848690, 80818650742133717, 1040037672241415947, 13829246515918840106

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Example

The compositions of 4 are [1,1,1,1], [2,1,1], [1,2,1], [1,1,2], [2,2], [3,1], [1,3], [4] giving the a(4) = 22 sequences: 1234, 1123, 1213, 1231, 1223, 2123, 1232, 1233, 1323, 3123, 1122, 1212, 1221, 2112, 2121, 1112, 1121, 1211, 1222, 2122, 2212, 1111.

Maple

c:= l-> f(l)*nops(l)!/(v-> mul(coeff(v, x, j)!,
        j=0..degree(v)))(add(x^i, i=l)):
g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
      binomial(n-1, l[-1]-1)+ add(f(sort(subsop(j=l[j]
      -1, l))), j=1..nops(l)-1))(add(i, i=l))
    end:
f:= l-> (n-> `if`(n<2 or l[-1]=1, 1, `if`(l[1]=0, 0, `if`(
         n=2, binomial(l[1]+l[2], l[1])-1, g(l)))))(nops(l)):
h:= (n, i, l)-> `if`(n=0 or i=1, c([1$n, l[]]), h(n, i-1, l)+
                `if`(i>n, 0, h(n-i, i, [i, l[]]))):
a:= n-> h(n$2, []):
seq(a(n), n=0..25);

Mathematica

c[l_] := f[l]*Length[l]!/Function[v, Product[Coefficient[v, x, j]!, {j, 0, Exponent[v, x]}]][Sum[x^i, {i, l}]];
g[l_] := g[l] = Function[n, f[Most@l]*Binomial[n-1, l[[-1]]-1] + Sum[f[ Sort[ReplacePart[l, j -> l[[j]]-1]]], {j, 1, Length[l]-1}]][ Total[l]];
f[l_] := Function[n, If[n < 2 || l[[-1]] == 1, 1, If[l[[1]] == 0, 0, If[n == 2, Binomial[l[[1]] + l[[2]], l[[1]]]-1, g[l]]]]][Length[l]];
h[n_, i_, l_] := If[n == 0 || i == 1, c[Join[Array[1&, n], l]], h[n, i-1, l] + If[i > n, 0, h[n-i, i, Join[{i}, l]]]];
a[n_] := h[n, n, {}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 06 2022, after Alois P. Heinz *)

Crossrefs

Cf. A047909, A268698, A268700, A268701.

Sequence in context: A087959 A292096 A006871 * A253948 A248836 A351919

Adjacent sequences: A268696 A268697 A268698 * A268700 A268701 A268702

Keyword

nonn

Author

Alois P. Heinz, Feb 11 2016

Status

approved