A337274 Number of distinct graceful labelings of trees with n vertices.

1, 1, 1, 2, 6, 20, 82, 376, 2010, 11788, 77816, 556016, 4366814, 36773666, 335394762, 3251474116, 33770466316, 370474881290, 4317182375632, 52861107601060, 683129289079532, 9234228045432682, 131059243976153410

Offset

1,4

Comments

Consider vertices numbered 1 to n. Add the edges 1--n, 2--n, and either 1--(n+1-k), 2--(n+2-k), ... or k--n for 3<=k<n. (Altogether (n-1)!/2 possibilities.) If the resulting graph has no cycles, this is a graceful labeling of a tree.

Links

Table of n, a(n) for n=1..23.

David Anick, Counting graceful labelings of trees: a theoretical and empirical study, Discrete Applied Mathematics 198 (2016), 65-81.

Formula

If n>3, a(n) = A033472(n)/2.

Example

For example, the six labelings for n=5 are:
  1--5, 2--5, 1--3, 3--4;
  1--5, 2--5, 1--3, 4--5;
  1--5, 2--5, 2--4, 2--3;
  1--5, 2--5, 2--4, 3--4;
  1--5, 2--5, 3--5, 3--4;
  1--5, 2--5, 3--5, 4--5.
There are three trees (A000055); the path P4 has two labelings, the graph K_{1,4} has one, the other ("chair" or "fork") has three.

Prog

(PARI) \\ After David Anick's C program GLs
a337274(N) = {  if(N<4, return(1)); my (n=N-1, count, i, j, k, m, nn, n1, u, v, z, a=vectorsmall(N), r= vectorsmall(N), t=vectorsmall(N), w=vectorsmall(N, i, -1));
  nn = n-1; n1 = n+1; w[n1] = 1; w[n] = 2; t[1] = n; t[2] = nn; m = nn - 1;
  while (a[nn]==0,
    for (jj=n1-m, n,
      u = a[n1-jj]; v = u + n1 - jj;
      z = w[u+1]; while (z > 0, u = r[z]; z = w[u+1]);
      z = w[v+1]; while (z > 0, v = r[z]; z = w[v+1]);
      if (v == u, j=jj; break);
      if (v > u, w[v+1] = jj; r[jj] = u; t[jj] = v,
                 w[u+1] = jj; r[jj] = v; t[jj] = u);
      j=jj+1
    );
    if (j > n, count++; w[t[n]+1] = -1 );
    if (j >= n, a[1]++; if (a[1] < nn, m = 1; next );
                a[1] = 0; j = nn; w[t[nn]+1] = -1 );
    m = n1 - j; a[m]++;
    while (a[m]> n-m, a[m]=0; a[m++]++; w[t[n1-m]+1]=-1)
  ); count};
for(k=1, 12, print1(a337274(k), ", ")) \\ Hugo Pfoertner, Sep 04 2020

Crossrefs

Cf. A000055, A033472.

Sequence in context: A177475 A118476 A260788 * A115084 A177476 A266600

Adjacent sequences: A337271 A337272 A337273 * A337275 A337276 A337277

Keyword

nonn,more

Author

Don Knuth, Sep 03 2020

Extensions

a(18)-a(19) from Bert Dobbelaere, Sep 06 2020

a(20)-a(23) (from A033472) from Joerg Arndt, Sep 15 2020

Status

approved