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

2,2

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)|.

Links

Table of n, a(n) for n=2..67.

Example

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
  ...

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 *)

Crossrefs

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

Sequence in context: A271684 A194682 A274105 * A056242 A343960 A128718

Adjacent sequences: A366153 A366154 A366155 * A366157 A366158 A366159

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 03 2023

Status

approved