A306800 Square array whose entry A(n,k) is the number of endofunctions on a set of size n with preimage constraint {0,1,...,k}, for n >= 0, k >= 0, read by descending antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 6, 0, 1, 1, 4, 24, 24, 0, 1, 1, 4, 27, 204, 120, 0, 1, 1, 4, 27, 252, 2220, 720, 0, 1, 1, 4, 27, 256, 3020, 29520, 5040, 0, 1, 1, 4, 27, 256, 3120, 44220, 463680, 40320, 0, 1, 1, 4, 27, 256, 3125, 46470, 765030, 8401680, 362880, 0

Offset: 0

Author

Benjamin Otto, Mar 10 2019

Keywords

• easy
• nonn
• tabl

Comments

A preimage constraint is a set of nonnegative integers such that the size of the inverse image of any element is one of the values in that set.

Thus, A(n,k) is the number of endofunctions on a set of size n such that each preimage has at most k entries. Equivalently, A(n,k) is the number of n-letter words from an n-letter alphabet such that no letter appears more than k times.

Examples

			Array begins:
  1    1     1     1     1 ...
  0    1     1     1     1 ...
  0    2     4     4     4 ...
  0    6    24    27    27 ...
  0   24   204   252   256 ...
  0  120  2220  3020  3120 ...
  0  720 29520 44220 46470 ...
  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.6 and 7.8.

Crossrefs

Columns 0-9: A000007, A000142, A012244, A239368, A324804, A324805, A324806, A324807, A324808, A324809.

A(n,n) gives A000312.

Similar array for preimage condition {i>=0 | i!=k}: A245413.

Number of functions with preimage condition given by the even nonnegative integers: A209289.

Sum over all k of the number of functions with preimage condition {0,k}: A231812.

Cf. A019575.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1, k)*binomial(n, j), j=0..min(k, n))))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 05 2019

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0 && i==0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] Binomial[n, j], {j, 0, Min[k, n]}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 29 2019, after Alois P. Heinz *)

Python

# print first num_entries entries in column k
import math, sympy; x=sympy.symbols('x')
k=5; num_entries = 64
P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [1]; curr_pow = 1
for term in range(1,num_entries):
   curr_pow=(curr_pow*eP).expand()
   r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)

Formula

A(n,k) = n! * [x^n] e_k(x)^n, where e_k(x) is the truncated exponential 1 + x + x^2/2! + ... + x^k/k!. When k>1, the link above yields explicit constants c_k, r_k so that the columns are asymptotically c_k * n^(-1/2) * r_k^-n. Stirling's approximation gives column k=1, and column k=0 is 0.

A(n,k) = Sum_{j=1..min(k,n)} A019575(n,j) for n>=1. - Alois P. Heinz, Jun 28 2023

Extensions

Offset changed to 0 by Alois P. Heinz, Jun 28 2023