A394503 Number of equivalence classes of set partitions of [n] under the adjacency monomial E.

1, 1, 2, 4, 10, 27, 79, 276, 950, 3943, 15865, 73882, 331200, 1725730, 8719631, 48170067, 266114734

Offset

0,3

Comments

Let s be a set partition of [n] having m blocks b_1,b_2,...,b_m with block b_m = {e_1,e_2,...,e_k} and e_1 < e_2 < ... < e_k. Then the adjacency monomial E associated with s equals Product_{b_m in s} Product_{e_i,e_j in b_m; i<j} x_{e_j - e_i}. Two set partitions of [n] are then considered equivalent if they have the same adjacency monomial E.

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = A381349(n,n) for n > 0.

Sum_{i=1..n} a(i) = Sum_{k=1..n} A381349(n,k) for n > 0.

Example

The set partition of [9]: {1,4,5},{2,6},{3},{7,8,9} has adjacency monomial E = (x_1*x_3*x_4) * (x_4) * (x^0) * (x_1^2*x_2) = x_1^3*x_2*x_3*x_4^2 abbreviated as the tuple (3,1,1,2).
a(4) = 10 counts the following monomials shown as tuples: (3,2,1), (2,1), (1,1,1), (2), (1,0,1), (0,1), (0,0,1), (0,2), (1), ().

Prog

(Python)
from more_itertools import set_partitions
from itertools import combinations
def adj_E(s):
    E = [0]
    for i in s:
        for j in combinations(i, 2):
            x = j[1] - j[0]
            while len(E) <= x: E.append(0)
            E[x] += 1
    return tuple(E)
def A394503(n):
    if n < 2: return 1
    return len(set(adj_E(i) for i in set_partitions(range(1, n+1))))

Crossrefs

Main diagonal of A381349.

Cf. A000110, A087903, A392756, A394235.

Sequence in context: A127386 A279551 A148106 * A291210 A340635 A099950

Adjacent sequences: A394499 A394500 A394502 * A394504 A394505 A394506

Keyword

nonn,more

Author

John Tyler Rascoe, Apr 30 2026

Extensions

a(15)-a(16) from Christian Sievers, May 19 2026

Status

approved