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A259976
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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).
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2
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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
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OFFSET
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0,13
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COMMENTS
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See Merris and Watkins (1983) for precise definition.
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LINKS
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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).
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FORMULA
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T(n,k) = A005368(k) for n >= 2*k. (End)
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EXAMPLE
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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
...
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PROG
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(Sage)
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)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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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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STATUS
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approved
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