cp's OEIS Frontend

The sum of the elements in the m-th row gives the total number of lattice paths from the origin to (m,m,m), and the number of elements in the m-th row is 3m^2+1. The (m,n)-th entry gives the number of lattice paths from the origin to (m,m,m) with inversion number n. The inversion number of a given lattice path depends on the ordering of the coordinates, but the total number of lattice paths with inversion number n does not. To calculate the inversion number w.r.t. an ordering of coordinates, fix such an ordered set of coordinates (x_1, x_2, x_3) and represent a lattice path, L, as a sequence, S(L), of m copies of each of the numbers {1,2,3} in the natural way (a step in the x_1 direction corresponds to a 1, a step in the x_2 direction corresponds to a 2, etc.).

(1) There is a natural way to associate each sequence, S(L), to a set of two partitions, P_1 and P_2, where P_1 fits in an box of size m x m and P_2 fits in a box of size m x 2m. The inversion number of L is given by |P_1|+|P_2|. See link.

(2) Equivalently, the inversion number of L is given by summing over each entry in S(L), the number of entries which precede that entry and whose value exceeds that entry’s.

The sequence is palindromic and unimodal by row.

Also, for n > 0, the number of totally symmetric (n-1)-dimensional partitions which fit in an (n-1)-dimensional box whose sides all have length 5.

There is no conjectured formula for a(n).

The formula a(n,d) = Product_{i_1=1..n} Product_{i_2=i_1..n} ... Product_{i_d=i_(d-1)..n} (i_1+i_2+...+i_d-d+2)/(i_1+i_2+...+i_d-d+1) gives the number of totally symmetric d-dimensional partitions that fit in a box whose sides all have length n, for d = 1, 2, and 3. For d > 3 this formula fails. In particular, when d=4 it produces the sequence: 1, 2, 6, 32, 352, 9216, 661504, ... rather than the sequence above.

a(n) is the sum over d from 1 to infinity of the number of totally symmetric d-dimensional Ferrers diagrams with n nodes.

A d-dimensional Ferrers diagram is totally symmetric if and only if whenever X=(x1,x2,...,xd) is a node, then so are all nodes which can be specified by permuting the coordinates of X.

Since a(1)=oo, the sequence above begins on n=2. All other terms are finite.

We can think of the points of a totally symmetric partition of n, say p, as occurring in classes, where two points are in the same class iff one point is a given by a permutation of the coordinates of the other. Call the number of distinct points in a class the size of that class.

The only classes of points in a 6-dimensional totally symmetric partition, p, of n, which do not have class size divisible by 3 are composed of points of the form (x,x,x,x,x,x) or (x,x,x,y,y,y) (or any permutation of these coordinates). The former has class size 1, the latter, class size 20.

For n=2 mod 3, a(n)=0 for the first 232 terms. Indeed, suppose n<233 and n=2 mod 3 and p partitions n in 6 dimensions. If j is the number of points of the form (x,x,x,x,x,x) in p, and k is the number of points of the form (x,x,x,y,y,y) in p, then we must have j+2k = 2 mod 3. Now j>0 because (1,1,1,1,1,1) must be a point of p. If j=1, we have k=2 mod 3, so that k>=2. In this case, the minimum size of n occurs when k=2 and the two points of the form (x,x,x,y,y,y) are (2,2,2,1,1,1) and (3,3,3,1,1,1). In this case, n=233. If j=2, we have k=0 mod 3. But since j=2,(2,2,2,2,2,2) is a point of p. Thus, so is(2,2,2,1,1,1). Hence, k>0, whence k>=4. In particular, k>=2 so that n>233. If j>=3, then (3,3,3,3,3,3) is a point of p, in which case n>729=3^6.

In fact the first term of the sequence with n=2 mod 3, and which is nonzero is a(233) = 1

a(n) gives the number of 5-dimensional Ferrers diagrams that have the property that if the point X=(x1, x2, x3, x4, x5) appears in the diagram, then so do all the points specified by the permutations of the coordinates of X.

We can think of the points of a totally symmetric partition of n, say p, as occurring in classes, where two points are in the same class iff one point is a given by a permutation of the coordinates of the other.

Suppose p is a 5-dimensional totally symmetric partition of n. For any point of n, say x = (x1, x2, x3, x4, x5), then, because 5 is prime, 5 divides the number of distinct permutations of the coordinates of x unless x1 = x2 = x3 = x4 = x5 (in which case there is only 1 such distinct permutation). Therefore, the only classes of points in p which have a number of points not divisible by 5 are points of the form (x,x,x,x,x). Hence, the number of points in p is equal to m mod 5, where m is the number of diagonal points, or points of the form (x,x,x,x,x), in p.

If 0 < n < 32=2^5, then the number of diagonal points in any 5-dimensional partition of n must be less than 2 (and greater than 0)—therefore equal to 1. Thus, for n < 32, a(n) is nonzero only if n=1 mod 5. Further, if 0 < n < 243=3^5, then the number of diagonal points in any 5-dimensional partition of n must be less than 3, thus equal to 1 or 2. Thus for n < 243, a(n) is nonzero only if n=1 mod 5 or n=2 mod 5. Consequently for n=0, 3, or 4 mod 5, a(n)=0 in the first 125 terms given above. A similar pattern occurs in a sequence of totally symmetric d-dimensional partitions of n whenever d is prime.

Generalization of the number of standard Young tableaux on a given Young diagram to arbitrary dimension.

The multidimensional Young numbers of partitions which are conjugate are equal. Therefore, the multidimensional Young numbers listed above are indexed with respect to an ordering of the "conjugacy classes" of partitions. This ordering is defined in the attached pdf file.

The number of entries between the m-th and (m+1)-th appearance of 1 (including the m-th appearance, but excluding the (m+1)-th) is the number of infinite dimensional partitions of m up to conjugacy, i.e., sequence A119268.

Let f(m) give the number of 2-dimensional partitions of m up to conjugacy (sequence A005987). Then the first f(m) entries following (and including) the m-th appearance of 1 are standard Young tableaux numbers on 2-dimensional partitions of m, and can be found in sequence A117506.