A347487 -id:A347487 - cp's OEIS frontend

A347842 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_4)^n.

1, 1, 6, 148, 14808, 5942608, 9546508128, 61355108818240, 1577381936031968640, 162213856617581098030336, 66726795842176170072717129216, 109792555585903911536355551233758208, 722612693482570097701467493432061846673408, 19023844570798442009810731239392846416136188284928

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 148 = 1 * 1 + 21 * 2 + 105 * 1, counting:
the unrefined chain 0 < (F_4)^3;
21 chains 0 < V < (F_4)^3, with dim(V) = 1; another
21 chains 0 < V < (F_4)^3, with dim(V) = 2; and
105 chains 0 < V_1 < V_2 < (F_4)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..58 (terms n = 1..40 from Álvar Ibeas)

Cf. A289545, A347487, A036038, A381299.

Column k=4 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(4^(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} A347487(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)} 4^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

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

A348115 Irregular triangle read by rows: T(n, k) is the number of chains of subspaces 0 < V_1 < ... < V_r = (F_4)^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, 7, 24, 1, 12, 31, 117, 469, 1, 19, 111, 458, 1435, 6356, 28753, 1, 29, 361, 964, 1579, 15266, 55470, 71660, 264300, 1267174, 6105030, 1, 41, 1068, 8042, 4886, 145628, 494779, 1952843, 705790, 9589197, 38323695, 47157299, 188963325, 932529235

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 105 (= 21 * 5) = A347487(3, 3) subspace chains 0 < V_1 < V_2 < (F_4)^3.
The permutations of the three coordinates classify them into 24 = T(3, 3) orbits.
T(3, 2) = 7 refers to partition (2, 1) and counts subspace chains in (F_4)^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  7  24
n=4:  1 12  31 117  469
n=5:  1 19 111 458 1435 6356 28753

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. A347971, A347487.

If the k-th partition of n in A-St is L = (a, n-a), then T(n, k) = A347971(n, a) = A347971(n, n-a).

cp's OEIS Frontend

A347842 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_4)^n.

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