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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.
11
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)|.
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Oct 03 2023
STATUS
approved