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A268701 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 into distinct parts. 4
1, 1, 1, 5, 7, 27, 195, 421, 1619, 8675, 105757, 274029, 1402193, 6625987, 55349787, 975068069, 3137395939, 17960895375, 101880880461, 696011551909, 7596647200175, 197122787505191, 723879298052695, 4905597865756059, 29537689035766501, 227793692735075911 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..90

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 into distinct parts are [3,1], [1,3], [4] giving the a(4) = 7 sequences: 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>i*(i+1)/2, 0, `if`(n=0, c(l), h(n, i-1, l)

               +`if`(i>n, 0, h(n-i, i-1, [i, l[]])))):

a:= n-> h(n$2, []):

seq(a(n), n=0..30);

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 > i (i + 1)/2, 0, If[n == 0, c[l], h[n, i - 1, l] + If[i > n, 0, h[n - i, i - 1, Join[{i}, l]]]]];

a[n_] := h[n, n, {}];

Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Jul 12 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A047909, A268698, A268699, A268700.

Sequence in context: A278646 A029668 A144392 * A317166 A126888 A185302

Adjacent sequences:  A268698 A268699 A268700 * A268702 A268703 A268704

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 11 2016

STATUS

approved

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Last modified October 2 08:34 EDT 2022. Contains 357191 sequences. (Running on oeis4.)