A366156 Triangular array read by rows: T(n,k) = number of pairs u,v of partitions of n such that d(u,v) = 2k, where d is the distance function defined in Comments.

1, 2, 1, 5, 4, 1, 9, 7, 4, 1, 17, 20, 11, 6, 1, 28, 35, 22, 13, 6, 1, 47, 70, 53, 35, 17, 8, 1, 73, 119, 104, 68, 41, 21, 8, 1, 114, 211, 197, 158, 87, 58, 25, 10, 1, 170, 337, 349, 282, 185, 111, 66, 29, 10, 1, 253, 555, 626, 560, 385, 267, 143, 89, 35, 12, 1

Offset: 2

Author

Clark Kimberling, Oct 03 2023

Keywords

• nonn
• tabl

Comments

Suppose that p = [p(1),...,p(i)] and q = [q(1),...,q(j)] are partitions of n, where p(1) >= ... >= p(i) and q(1) >= ... >= q(j). If i = n, let p_ = p, else p_ = [p(1),...,p(i),0,...,0], where the number of 0' s appended is n-i. If j = n, let q_ = q, else q_ = [q(1),...,q(j),0,...,0], where the number of 0's appended is n-j. Write p_ = [p(1),...,p(i),p(i+1),...,p(n)] and q_ = [q(1),...,q(j),q(j+1),...,q(n)]. The distance between p and q is defined by d(p,q) = |p(1) - q(1)| + ... + |p(n) - q(n)|.

Examples

			Write the 5 partitions of 4 as 4, 31, 22, 211, 111, and represent them as a,b,c,d,e in the following tableaux:
 a : 4 0 0 0  | 2 4 4 6
 b : 3 1 0 0  | 2 2 4
 c : 2 2 0 0  | 2 4
 d : 2 1 1 0  | 2
 e : 1 1 1 1
where, for example, the distances 2 4 4 6 are given by
 d(a,b) = |4-3| + |0-1| + |0-0| + |0-0| = 2
 d(a,c) = |4-2| + |0-2| + |0-0| + |0-0| = 4
 d(a,d) = |4-2| + |0-1| + |0-1| + |0-0| = 4
 d(a,e) = |4-1| + |0-1| + |0-1| + |0-1| = 6
First eight rows:
   1
   2     1
   5     4     1
   9     7     4    1
  17    20    11    6    1
  28    35    22   13    6    1
  47    70    53   35   17    8   1
  73   119   104   68   41   21   8   1
  ...
		

Crossrefs

Cf. A000041, A000097 (column 1), A230025 (see Comment), A355389 (row sums).

Programs

Mathematica

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
t[n_] := Flatten[Table[d[r[n, j], r[n, k]], {j, 1, -1 + c[n]}, {k, j + 1, c[n]}]];
t1 = Table[Count[t[n], m], {n, 2, 17}, {m, 2, 2 n - 2, 2}]
TableForm[t1] (* this sequence as an array *)
u = Flatten[t1]  (* this sequence *)