A001372 -id:A001372 - cp's OEIS frontend

A001373 Number of functional digraphs (digraphs of functions on n nodes where every node has outdegree 1 and loops of length 1 are forbidden).

1, 0, 1, 2, 6, 13, 40, 100, 291, 797, 2273, 6389, 18264, 51916, 148666, 425529, 1221900, 3511507, 10111043, 29142941, 84112009, 243000149, 702758065, 2034150215, 5892907566, 17084615940, 49567063847, 143902155133, 418032946298, 1215076634226, 3533715961160, 10282042126394, 29931877173282, 87173224346464, 253989569994664

Offset: 0

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 70, Table 3.4.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joerg Arndt, Table of n, a(n) for n = 0..508
Frank Harary, The number of functional digraphs, Mathematische Annalen, 138.3 (1959): 203-210. - From _N. J. A. Sloane_, Apr 08 2014
Ronald C. Read, Note on number of functional digraphs, Math. Ann., vol. 143 (1961), pp. 109-111.
N. J. A. Sloane, Transforms
L. Travis, Graphical Enumeration: A Species-Theoretic Approach, arXiv:math/9811127 [math.CO], 1998.
James Turner and William H. Kautz, A survey of progress in graph theory in the Soviet Union, SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 20 concerning calculation of a(n) for n <= 25 by Zakrevskii in 1961. - _N. J. A. Sloane_, Apr 08 2014

Column k=1 of A329228.

Cf. A001372, A002862, A000081.

Needs["Combinatorica`];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,2,30}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]  (* after code given by Robert A. Russell in A000081 *) (* Geoffrey Critzer, Oct 12 2012 *)

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1,n, sumdiv(k,d, d*A[d]) * A[n-k+1] ) );
v0000081=concat([0], A); \\ A000081
x='x+O('x^N);  T = Ser(v0000081);
gf = x/T  / prod(n=1,N, 1 - subst(T,'x,'x^n) );
v001373 = Vec(gf) \\ Joerg Arndt, Apr 17 2014

Euler transform of A002862.

G.f.: (x/T(x)) / Product_{n>=1} ( 1 - T(x^n) ) where T(x) is the g.f. of A000081, see the Read reference and the PARI code. - Joerg Arndt, Apr 17 2014

Sequence extended by Paul Zimmermann
More terms and better description from Christian G. Bower
More terms added by Joerg Arndt, Apr 17 2014

A054745 Number of nonisomorphic binary n-state automata without output under input permutations.

Original entry on oeis.org

1, 1, 7, 74, 1474, 41876, 1540696, 68343112, 3540691525, 209612916303, 13957423192794, 1032436318269648, 83993175608894096, 7453446303042245261, 716451740543945788671, 74159075140708644544128, 8223831291824019614386868, 972718473204236819072891710

Offset: 0

Vladeta Jovovic, Apr 22 2000

nonn

Also isomorphism classes of unordered pairs of endofunctions i.e. an unorder pair {f,g} of functions from {1,...,n} to itself. - Christian G. Bower, Dec 18 2003

F. Harary and E. Palmer, Graphical Enumeration, 1973.

Alois P. Heinz, Table of n, a(n) for n = 0..45
M. A. Harrison, A census of finite automata, Canad. J. Math., 17, No. 1, 1965, p. 110.

Cf. A001372, A054050, A054732, A054746.

with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
a:= proc(n) option remember; add(add(mul(mul(add(coeff(s, x, d)
      *d, d=divisors(ilcm(i, j)))^(igcd(i, j)*coeff(s, x, i)*
      coeff(t, x, j)), j=1..degree(t)), i=1..degree(s))
      /mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t))
      /mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))
      , t=b(2$2)), s=b(n$2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2014

b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Table[Map[Function[{p}, p + j*x^i], b[n - i*j, i-1]], {j, 0, n/i}] // Flatten // Union]];
a[n_] := a[n] = Sum[Sum[Product[Product[With[{g = GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j]}, If[g==0, 1, Sum[Coefficient[s, x, d]*d, {d, Divisors[LCM[i, j]]}]^g]], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/
Product[i^Coefficient[t, x, i]Coefficient[t, x, i]!, {i, Exponent[t, x]}]/
Product[i^Coefficient[s, x, i]Coefficient[s, x, i]!, {i, Exponent[s, x]}], {t, b[2, 2]}], {s, b[n, n]}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz, updated Jan 01 2021 *)

a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2=2} (fixA[s_1, s_2, ...;t_1, t_2]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!)) where fixA[...] = prod {i, j>=1} ( (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*t_j)). - Christian G. Bower, Dec 18 2003

More terms from Alois P. Heinz, Aug 15 2014

A054050 Number of nonisomorphic binary n-state automata.

Original entry on oeis.org

1, 10, 129, 2836, 83061, 3076386, 136647824, 7081061404, 419223006090, 27914819962058, 2064872379041701, 167986348586006675, 14906892578198245332, 1432903480780688968334, 148318150277923875087238, 16447662583606982784243526, 1945436946407977282783367806, 244476687257496605058725664611

Offset: 1

Vladeta Jovovic, Apr 29 2000

nonn

Also, number of isomorphism classes of ordered pairs of endofunctions (i.e., an order pair (f,g) of functions) from {1,...,n} to itself. - Christian G. Bower, Dec 18 2003

			From _Petros Hadjicostas_, Mar 08 2021: (Start)
For n = 2, we have the partitions 1*2 + 2*0 and 1*0 + 2*1 of 2 (in frequency notation). The corresponding denominators in _Christian G. Bower_'s formula are 1^2*2!*2^0*0! = 2 and 1^0*0!*2^1*1! = 2.
Also, fixA[s_1 = 2, s_2 = 0] = (Sum_{d|1} d*s_d)^(2*s_1) * (Sum_{d|2} d*s_d)^(2*s_2) = (1*s_1)^(2*s_1) = 16. In addition, fixA[s_1 = 0, s_2 = 1] = (Sum_{d|1} d*s_d)^(2*s_1) * (Sum_{d|2} d*s_d)^(2*s_2) = (1*s_1 + 2*s_2)^(2*s_2) = (0 + 2)^2 = 4. Hence a(2) = 16/2 + 4/2 = 10. (End)

F. Harary and E. Palmer, Graphical Enumeration, 1973.

M. A. Harrison, A census of finite automata, Canad. J. Math., 17, No. 1, (1965), 100-113. [See p. 107 (Theorem 6.1 with k = 2 and p = 1) and p. 110 (Table I).]

Cf. A001372, A002854, A054745, A054051.

A054050(n)={local(p=vector(n));
my(S=0, A() = prod(i=1, n, sumdiv(i, d, d*p[d])^(2*p[i])),
inc()=!forstep(i=n, 1, -1, p[i]n, p[i]=n); next(2))); t==n && S+ = A()/prod(i=1, n, i^p[i]*p[i]!)); S} \\ This is a modification of M. F. Hasler's PARI program from A002854. - Petros Hadjicostas, Mar 08 2021

a(n) = Sum_{1*s_1+2*s_2+...=n} fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...), where fixA[s_1, s_2, ...] = Product_{i>=1} (Sum_{d|i} d*s_d)^(2*s_i). - Christian G. Bower, Dec 18 2003 [Corrected by Petros Hadjicostas, Mar 08 2021 using Theorem 6.1 in Harrison (1965) with k = 2 inputs and p = 1 output.]

a(16)-a(18) from Petros Hadjicostas, Mar 08 2021

A126285 Number of partial mappings (or mapping patterns) from n points to themselves; number of partial endofunctions.

Original entry on oeis.org

1, 2, 6, 16, 45, 121, 338, 929, 2598, 7261, 20453, 57738, 163799, 465778, 1328697, 3798473, 10883314, 31237935, 89812975, 258595806, 745563123, 2152093734, 6218854285, 17988163439, 52078267380, 150899028305, 437571778542, 1269754686051, 3687025215421

Offset: 0

Christian G. Bower, Dec 25 2006 based on a suggestion from Jonathan Vos Post

nonn

If an endofunction is partial, then some points may be unmapped (or mapped to "undefined").

The labeled version is left-shifted A000169. - Franklin T. Adams-Watters, Jan 16 2007

Alois P. Heinz, Table of n, a(n) for n = 0..750
R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier, What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations, arXiv preprint arXiv:1512.00906 [math.NA], 2015-2017.
N. J. A. Sloane, Transforms

Cf. A001372.

Cf. A000169, A000081, A002861.

nmax = 28;
a81[n_] := a81[n] = If[n<2, n, Sum[Sum[d*a81[d], {d, Divisors[j]}]*a81[n-j ], {j, 1, n-1}]/(n-1)];
A[n_] := A[n] = If[n<2, n, Sum[DivisorSum[j, #*A[#]&]*A[n-j], {j, 1, n-1} ]/(n-1)];
H[t_] := Sum[A[n]*t^n, {n, 0, nmax+2}];
F = 1/Product[1 - H[x^n], {n, 1, nmax+2}] + O[x]^(nmax+2);
A1372 = CoefficientList[F, x];
a[n_] := Sum[a81[k] * A1372[[n-k+2]], {k, 0, n+1}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 18 2018, after Franklin T. Adams-Watters *)

Pol. = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
    if n==0: return Pol.one()
    return sum(t[k]*Z(n-k) for k in (1..n))/n
def pmagmas(n,k=1): # number of partial k-magmas on a set of n elements up to isomorphism
    P = Z(n)
    q = 0
    coeffs = P.coefficients()
    count = 0
    for m in P.monomials():
        p = 1
        V = m.variables()
        T = cartesian_product(k*[V])
        for t in T:
            r = [Pol.varname_key(str(u))[1] for u in t]
            j = [m.degree(u) for u in t]
            D = 1
            lcm_r = lcm(r)
            for d in divisors(lcm_r):
                try: D += d*m.degrees()[-d-1]
                except: break
            p *= D^(prod(r)/lcm_r*prod(j))
        q += coeffs[count]*p
        count += 1
    return q
# Philip Turecek, Nov 27 2023

Euler transform of A002861 + A000081 = [1, 2, 4, 9, 20, 51, 125, 329, 862, 2311, ... ] + [ 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, ...] = A124682.
Convolution of left-shifted A000081 with A001372. - Franklin T. Adams-Watters, Jan 16 2007
a(n) ~ c * d^n / sqrt(n), where d = 2.95576528565... is the Otter's rooted tree constant (see A051491) and c = 1.309039781943936352117502717... - Vaclav Kotesovec, Mar 29 2014

A116950 Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.

Original entry on oeis.org

1, 2, 7, 20, 61, 174, 514, 1478, 4303, 12437, 36084, 104494, 303167, 879283, 2552803, 7413583, 21544347, 62635823, 182199853, 530228946, 1543761513, 4496523995, 13102414665, 38193626823, 111375529695, 324891970936, 948051861938, 2767336312386, 8080206646244

Offset: 0

Franklin T. Adams-Watters, Mar 29 2006

A001372 counts functional patterns from a set with n elements to itself; A000041 (partition function) counts functional patterns from a set with n elements to a disjoint set; this is the general case where the range may overlap the domain but may also include other values.

			For n=2 there are the following 7 digraphs:
o-+.o-+ o->o-+ o->o o-+.o->o o->o->o o->o o->o
^.|.^.| ...^.| ^..| ^.|..... ....... ...^ ....
+-+.+-+ ...+-+ +--+ +-+..... ....... o--+ o->o

Alois P. Heinz, Table of n, a(n) for n = 0..2127

Cf. A000041, A001372, A002861, A000081.

nmax = 750;
A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt", "Table"], {, }][[;; nmax + 2, 2]];
A000081 = Cases[Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[;; nmax + 2, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
b[n_] := A002861[[n]] + A000081[[n + 2]];
a = etr[b];
a[0] = 1;
a /@ Range[0, nmax](* Jean-François Alcover, Mar 13 2020 *)

Euler transform of A002861(n) + A000081(n+1).

a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.95576528565199497471481752412..., c = 3.435908969217935496995961718... . - Vaclav Kotesovec, Sep 10 2014

A125024 Minimal Goedel number of endofunctions on k points, by row and sorted within rows (number of points).

Original entry on oeis.org

1, 2, 6, 12, 18, 30, 60, 90, 150, 540, 1500, 2250, 210, 420, 630, 1050, 1890, 2520, 3150, 3780, 7560, 9450, 15750, 18900, 28350, 75600, 113400, 126000, 141750, 216000, 246960, 5402250, 2310, 4620, 6930, 11550

Offset: 0

Jonathan Vos Post, Nov 15 2006

If one writes an endofunction over n points (finite n) and numbers the points 1 through n, then each labeled sagittal graph of an endofunction from (1,2,3,..., z) to (a, b, c, ..., z) with a, b, c, ..., z elements of (1,2,3,..., z), is isomorphic to a Goedel number prime(1)^a * prime(2)^b * ... * prime(z)^z. Then, among all the Goedel numbers for different labeled endofunctions of the same (to isomorphism) unlabeled endofunction, there is a unique minimal Goedel number, which is thus the Goedel number for that unlabeled endofunction. Similarly, there is a minimal Goedel number for all the endofunctions on n points, for any n. Iterating the endofunction yields those x for which f^k(n) = x, allowing us to see the cycle index. The length of the n-th row is A001372(n). Each row begins with primorial n# = A002110(n) corresponding to the endofunction (1, 2, 3, ..., n) -> (1, 1, 1, ..., 1) which maps every integer other than 1 to 1 and loops 1 to itself; and ends with the Goedel number of the endofunction consisting only of a cycle of length n labeled (1, 2, 3, ..., n-1, n) -> (n-1, n-2, ..., n, 1). The sequence includes A074736(n) which corresponds to the identity endofunction on n points.

			For example, take the endofunction on 4 points which is composed of two 2-cycles. We can write this as: (1, 2, 3, 4) -> (2, 1, 4, 3) which has the Goedel number 2^2 * 3^1 * 5^4 * 7^3 = 2572500. We can also renumber the points (applying the symmetric group S_n: n^n -> n^n) and write it: (1, 2, 3, 4) -> (3, 4, 1, 2) which gives us the Goedel number 2^3 * 3*4 * 5^1 * 7^2 = 158760. But the minimal Goedel number for that endofunction comes from:
(1, 2, 3, 4) -> (4, 3, 2, 1) which gives us 756000. Hence I can enumerate all the minimal Goedel numbers for the 7 endofunctions on 4 points as: 30, 60, 90, 150, 540, 1500, 2250.
Table begins:
n | row(n) of Goedel numbers
0 | 1. (formally defining prime(0) = 1)
1 | 2.
2 | 6, 12, 18.
3 | 30, 60, 90, 150, 540, 1500, 2250.
4 | 210, 420, 630, 1050, 1890, 2520, 3150, 3780, 7560, 9450, 15750, 18900, 28350, 75600, 113400, 126000, 141750, 216000, 246960, 5402250.
5 | 2310, 4620, 6930, 11550, ...
6 | 30030, 60060, 90090, 150150, ...
7 | 510510, 1021020, 1531530, ...
8 | 9699690, 19399380.

Cf. A001372, A002110, A074736 Goedel encoding of the prime factors of n, in increasing order and repeated according to multiplicity, A076954 Product_{i=1..n} prime(i)^i, A125023 Number of mappings (or mapping patterns) from k =< n points to themselves; number of endofunctions on k <= n points.

a(A125023(n)) = A002110(n).

A259471 Triangle read by rows: T(n,k) is the number of semi-regular relations on n nodes with each node having out-degree k (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 19, 66, 19, 1, 1, 47, 916, 916, 47, 1, 1, 130, 16816, 91212, 16816, 130, 1, 1, 343, 373630, 12888450, 12888450, 373630, 343, 1, 1, 951, 9727010, 2411213698, 14334255100, 2411213698, 9727010, 951, 1, 1, 2615, 289374391, 575737451509, 22080097881081, 22080097881081, 575737451509, 289374391, 2615, 1

Offset: 0

N. J. A. Sloane, Jul 03 2015

			Triangle begins:
  1;
  1,   1;
  1,   3,     1;
  1,   7,     7,     1;
  1,  19,    66,    19,     1;
  1,  47,   916,   916,    47,   1;
  1, 130, 16816, 91212, 16816, 130, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
S. A. Choudum, K. R. Parthasarathy, Semi-regular relations and digraphs, Nederl. Akad. Wetensch. Proc. Ser. A. {75}=Indag. Math. 34 (1972), 326-334.

Columns k=1..3 are A001372, A003286, A005535.

Cf. A329228.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, k_] := Product[SeriesCoefficient[Product[g = GCD[v[[i]], v[[j]] ]; (1 + x^(v[[j]]/g) + O[x]^(k + 1))^g, {j, 1, Length[v]}], {x, 0, k}], {i, 1, Length[v]}];
T[n_, k_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, k], {p, IntegerPartitions[n]}]; s/n!];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v,k)={prod(i=1, #v, polcoef(prod(j=1, #v, my(g=gcd(v[i],v[j])); (1 + x^(v[j]/g) + O(x*x^k))^g), k))}
T(n,k)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,k)); s/n!} \\ Andrew Howroyd, Sep 13 2020

T(n,k) = T(n,n-k). - Andrew Howroyd, Sep 13 2020

Terms a(28) and beyond from Andrew Howroyd, Sep 13 2020

A362897 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of endofunctions on an n-set with k endofunctions.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 7, 1, 1, 1, 13, 74, 19, 1, 1, 1, 22, 638, 1474, 47, 1, 1, 1, 34, 4663, 118949, 41876, 130, 1, 1, 1, 50, 28529, 7643021, 42483668, 1540696, 343, 1, 1, 1, 70, 151600, 396979499, 33179970333, 23524514635, 68343112, 951, 1

Offset: 0

Andrew Howroyd, May 10 2023

Isomorphism is up to permutations of the elements of the n-set.

			Array begins:
======================================================================
n/k| 0   1       2           3               4                   5 ...
---+------------------------------------------------------------------
0  | 1   1       1           1               1                   1 ...
1  | 1   1       1           1               1                   1 ...
2  | 1   3       7          13              22                  34 ...
3  | 1   7      74         638            4663               28529 ...
4  | 1  19    1474      118949         7643021           396979499 ...
5  | 1  47   41876    42483668     33179970333      20762461502595 ...
6  | 1 130 1540696 23524514635 274252613077267 2559276179593762172 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..3 are A000012, A001372, A054745, A362898.
Row n=2 is A002623.
Main diagonal is A277839.
Cf. A362644.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(v,m) = {prod(i=1, #v, my(g=gcd(v[i],m), e=v[i]/g); sum(j=1, #v, my(t=v[j]); if(e%(t/gcd(t,m))==0, t))^g)}
T(n,k) = {if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(q,m)*x^m/m, O(x*x^k))), k)); s/n!)}

T(0,k) = T(1,k) = 1.

A254529 a(n) = n! * (number of mapping patterns on n).

Original entry on oeis.org

1, 1, 6, 42, 456, 5640, 93600, 1728720, 38344320, 948931200, 26555558400, 817935148800, 27735629644800, 1020596255078400, 40642432179148800, 1737890081351424000, 79498734605402112000, 3871319396080840704000, 200017645344178421760000, 10925549584125028909056000

Offset: 0

Martin Fuller, Feb 01 2015

a(n) is the number of ordered pairs (p, f) such that p f = f p, where p is a permutation and f is an endofunction.

Alois P. Heinz, Table of n, a(n) for n = 0..382
Math Overflow, What is the probability two random maps on n symbols commute?, 2013.

Cf. A001372, A053529, A181162.

a(n) = n! * A001372(n). - Joerg Arndt, Feb 01 2015

A362898 Number of nonisomorphic unordered triples of endofunctions on an n-set.

Original entry on oeis.org

1, 1, 13, 638, 118949, 42483668, 23524514635, 18477841853059, 19526400231564564, 26714255854638149145, 45938634690184528585855, 96990967579564469350287613, 246664446832722382952639028437, 743740463651636548809953508690270, 2623456914417867939801667337352637825

Offset: 0

Andrew Howroyd, May 10 2023

nonn

Isomorphism is up to permutation of the elements of the n-set.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k=3 of A362897.

Cf. A001372, A054745.

cp's OEIS Frontend

A001373 Number of functional digraphs (digraphs of functions on n nodes where every node has outdegree 1 and loops of length 1 are forbidden).

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A054745 Number of nonisomorphic binary n-state automata without output under input permutations.

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A054050 Number of nonisomorphic binary n-state automata.

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A126285 Number of partial mappings (or mapping patterns) from n points to themselves; number of partial endofunctions.

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A116950 Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.

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A125024 Minimal Goedel number of endofunctions on k points, by row and sorted within rows (number of points).

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A259471 Triangle read by rows: T(n,k) is the number of semi-regular relations on n nodes with each node having out-degree k (0 <= k <= n).

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