A347486 -id:A347486 - cp's OEIS frontend

A347841 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_3)^n.

1, 1, 5, 79, 3851, 567733, 251790113, 335313799327, 1340040415899803, 16067553466179577453, 577986341168068075687337, 62375143109859674070751394743, 20194282336027244435564571244298243, 19614041602745899032342581715038226919285

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 79 = 1 * 1 + 13 * 2 + 52 * 1, counting:
the unrefined chain 0 < (F_3)^3;
13 chains 0 < V < (F_3)^3, with dim(V) = 1; another
13 chains 0 < V < (F_3)^3, with dim(V) = 2; and
52 chains 0 < V_1 < V_2 < (F_3)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..64 (terms n=1..40 from Álvar Ibeas)

Cf. A289545, A347486, A036038, A381299.

Column k=3 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(3^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 21 2025

a(n) = Sum_{L partition of n} A347486(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(n) = Sum_{k=0..binomial(n,2)} 3^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

a(0)=1 prepended by Alois P. Heinz, Feb 21 2025

A348114 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_3)^n, counted up to coordinate permutation, with dimension increments given by (any fixed permutation of) the parts of the k-th partition of n in Abramowitz-Stegun order.

Original entry on oeis.org

1, 1, 3, 1, 5, 15, 1, 8, 16, 49, 154, 1, 11, 39, 126, 288, 964, 3275, 1, 15, 87, 168, 291, 1412, 3600, 4957, 12865, 46400, 168862, 1, 19, 176, 644, 608, 6101, 14001, 38996, 22294, 146064, 418072, 549894, 1586761, 6045724, 23115063, 1, 24, 338, 2348, 4849, 1195, 24329

Offset: 1

Álvar Ibeas, Oct 01 2021

A permutation on the list of dimension increments does not modify the number of subspace chains.

The length of the enumerated chains is r = len(L), where L is the parameter partition.

			For L = (1, 1, 1), there are 52 (= 13 * 4) = A347486(3, 3) subspace chains 0 < V_1 < V_2 < (F_3)^3.
The permutations of the three coordinates classify them into 15 = T(3, 3) orbits.
T(3, 2) = 5 refers to partition (2, 1) and counts subspace chains in (F_3)^2 with dimensions (0, 2, 3), i.e. 2-dimensional subspaces. It also counts chains with dimensions (0, 1, 3), i.e. 1-dimensional subspaces.
Triangle begins:
  k:  1  2  3   4   5   6    7
      ------------------------
n=1:  1
n=2:  1  3
n=3:  1  5 15
n=4:  1  8 16  49 154
n=5:  1 11 39 126 288 964 3275

Álvar Ibeas, Table of n, a(n) for n = 1..65
Álvar Ibeas, First 16 rows, with gaps
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-2, 1, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-3, 2, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-3, 1, 1, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 3, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 2, 2), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 2, 1, 1), up to n=100
Álvar Ibeas, Pseudo-column T(n, L), where L = (n-4, 1, 1, 1, 1), up to n=100

Cf. A347970, A347486.

If the k-th partition of n in A-St is L = (a, n-a), then T(n, k) = A347970(n, a) = A347970(n, n-a).

cp's OEIS Frontend

A347841 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_3)^n.

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