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A337274 Number of distinct graceful labelings of trees with n vertices. 4
1, 1, 1, 2, 6, 20, 82, 376, 2010, 11788, 77816, 556016, 4366814, 36773666, 335394762, 3251474116, 33770466316, 370474881290, 4317182375632, 52861107601060, 683129289079532, 9234228045432682, 131059243976153410 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 16 08:18 EDT 2021. Contains 347469 sequences. (Running on oeis4.)