A331509
Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set-systems on a k-set.
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 15, 19, 1, 0, 0, 1, 0, 1, 42, 141, 29, 0, 0, 0, 1, 0, 1, 109, 1571, 769, 23, 0, 0, 0, 1, 0, 1, 320
Offset: 0
Array begins:
=================================
n\k | 0 1 2 3 4 5 6 7
----+----------------------------
0 | 1 1 0 0 0 0 0 0 ...
1 | 1 1 1 1 1 1 1 1 ...
2 | 1 0 1 3 6 15 42 109 ...
3 | 1 0 0 3 19 141 1571 ...
4 | 1 0 0 1 29 769 ...
5 | 1 0 0 0 23 ...
...
The A(2,3) = 3 matrices are:
[1 1 1] [1 1 0] [1 1 0]
[1 0 0] [1 0 1] [1 0 1]
[0 1 0] [0 1 0] [0 1 1]
[0 0 1] [0 0 1]
A259976
Irregular triangle T(n, k) read by rows (n >= 0, 0 <= k <= A011848(n)): T(n, k) is the number of occurrences of the principal character in the restriction of xi_k to S_(n)^(2).
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 4, 6, 6, 3, 1, 0, 1, 3, 5, 11, 20, 24, 32, 34, 17, 1, 0, 1, 3, 6, 13, 32, 59, 106, 181, 261, 317, 332, 245, 89, 1, 0, 1, 3, 6, 14, 38, 85, 197, 426, 866, 1615, 2743, 4125, 5495, 6318, 6054, 4416, 1637
Offset: 0
The triangle begins:
[0] 1
[1] 1
[2] 1
[3] 1,0,
[4] 1,0,1,1,
[5] 1,0,1,2,2,0,
[6] 1,0,1,3,4,6,6,3,
[7] 1,0,1,3,5,11,20,24,32,34,17
[8] 1,0,1,3,6,13,32,59,106,181,261,317,332,245,89
[9] 1,0,1,3,6,14,38,85,197,426,866,1615,2743,4125,5495,6318,6054,4416,1637
...
- Russell Merris and William Watkins, Tensors and graphs, SIAM J. Algebraic Discrete Methods 4 (1983), no. 4, 534-547.
- Andrey Zabolotskiy, a259976 (implementation in Rust).
Row n is apparently formed by the first differences of the first half of row n of
A008406.
-
from sage.groups.perm_gps.permgroup_element import make_permgroup_element
for p in range(8):
m = p*(p-1)//2
Sm = SymmetricGroup(m)
denom = factorial(p)
elements = []
for perm in SymmetricGroup(p):
t = perm.tuple()
eperm = []
for v2 in range(p):
for v1 in range(v2):
w1, w2 = sorted([t[v1], t[v2]])
eperm.append((w2-1)*(w2-2)//2+w1)
elements.append(make_permgroup_element(Sm, eperm))
for q in range(m//2+1):
char = SymmetricGroupRepresentation([m-q, q]).to_character()
numer = sum(char(e) for e in elements)
print((p, q), numer//denom)
# Andrey Zabolotskiy, Aug 28 2018
Name edited, terms T(7, 9)-T(7, 10) and rows 0-2, 8, 9 added by
Andrey Zabolotskiy, Sep 06 2018
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