A394503 - OEIS

%I #24 May 19 2026 19:41:05

%S 1,1,2,4,10,27,79,276,950,3943,15865,73882,331200,1725730,8719631,

%T 48170067,266114734

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

%C 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.

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

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

%e 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).

%e 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), ().

%o (Python)

%o from more_itertools import set_partitions

%o from itertools import combinations

%o def adj_E(s):

%o     E = [0]

%o     for i in s:

%o         for j in combinations(i,2):

%o             x = j[1] - j[0]

%o             while len(E) <= x: E.append(0)

%o             E[x] += 1

%o     return tuple(E)

%o def A394503(n):

%o     if n < 2: return 1

%o     return len(set(adj_E(i) for i in set_partitions(range(1,n+1))))

%Y Main diagonal of A381349.

%Y Cf. A000110, A087903, A392756, A394235.

%K nonn,more

%O 0,3

%A _John Tyler Rascoe_, Apr 30 2026

%E a(15)-a(16) from _Christian Sievers_, May 19 2026