A393748 Number of totally symmetric planar subpartitions of the size n pyramidal planar partition.

1, 2, 3, 6, 15, 44, 125, 568, 2337, 12650, 78770, 611952, 4670612, 53985322, 588296122, 8476173572, 136159829764, 2789679854068, 56580675586853, 1695358864944894, 48760209017690780, 1861276478642614456, 77886241973352349297, 4206052280136144958052, 225888522329929627644590

Offset

0,2

Comments

The planar partition having its subpartitions counted is:

n n-1 ... 2 1

n-1 n-2 ... 1

... ...

2 1

1

A planar partition is said to be totally symmetric if it is invariant by the action of S_3.

The number of all subpartitions regardless of symmetry is given by A115965.

The number of symmetric subpartitions is given by A388969.

The number of cyclically symmetric subpartitions is given by A393786.

Links

Ludovic Schwob, Table of n, a(n) for n = 0..34

Example

The a(4) = 15 totally symmetric planar subpartitions of the pyramidal planar partition of size 4 are:
  Ø        1        21       22       22       311      321      321
                    1        21       22       1        21       22
                                               1        1        1
.
  332      332      4111     4211     4211     4321     4321
  311      321      1        2        22       311      321
  21       21       1        1        1        21       21
                    1        1        1        1        1

Prog

(Python)
def covers(l0, n, k):
    L = [[]]
    for i in range(min(k, (n-k-1)//2)+1):
        L2 = []
        b = l0[i] if i < len(l0) and l0[i] < k-i else k-i+1
        b = min(b, n-k-2*i)
        for l in L:
            if i > 0:
                b2 = min(l[i-1]-1, b) if l[i-1]>0 else 0
            else:
                b2 = b
            for j in range(0, b2+1):
                l2 = l.copy()
                l2.append(j)
                L2.append(l2)
        L = L2
    return [tuple(l) for l in L]
def a(n):
    t = tuple()
    D = {t:1}
    for k in range(n):
        D2 = {}
        for l in D:
            for t in covers(l, n, k):
                if t in D2:
                    D2[t] += D[l]
                else:
                    D2[t] = D[l]
        D = D2
    return sum(D.values())
print([a(n) for n in range(15)])

Crossrefs

Cf. A005157, A115965, A388969, A393786.

Sequence in context: A110181 A141351 A088793 * A322197 A368954 A216144

Adjacent sequences: A393745 A393746 A393747 * A393749 A393750 A393751

Keyword

nonn

Author

Ludovic Schwob, Feb 26 2026

Status

approved