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%I #30 Mar 07 2020 12:09:05
%S 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,
%T 34,17,1,0,1,3,6,13,32,59,106,181,261,317,332,245,89,1,0,1,3,6,14,38,
%U 85,197,426,866,1615,2743,4125,5495,6318,6054,4416,1637
%N 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).
%C See Merris and Watkins (1983) for precise definition.
%H Russell Merris and William Watkins, <a href="https://doi.org/10.1137/0604054">Tensors and graphs</a>, SIAM J. Algebraic Discrete Methods 4 (1983), no. 4, 534-547.
%H Andrey Zabolotskiy, <a href="https://github.com/colt-browning/a259976">a259976</a> (implementation in Rust).
%F From _Andrey Zabolotskiy_, Aug 28 2018: (Start)
%F Sum_{ k=0..A011848(n) } T(n,k) * (n*(n-1)/2 - 2*k + 1) = A000088(n).
%F T(n,k) = A005368(k) for n >= 2*k. (End)
%e The triangle begins:
%e [0] 1
%e [1] 1
%e [2] 1
%e [3] 1,0,
%e [4] 1,0,1,1,
%e [5] 1,0,1,2,2,0,
%e [6] 1,0,1,3,4,6,6,3,
%e [7] 1,0,1,3,5,11,20,24,32,34,17
%e [8] 1,0,1,3,6,13,32,59,106,181,261,317,332,245,89
%e [9] 1,0,1,3,6,14,38,85,197,426,866,1615,2743,4125,5495,6318,6054,4416,1637
%e ...
%o (Sage)
%o from sage.groups.perm_gps.permgroup_element import make_permgroup_element
%o for p in range(8):
%o m = p*(p-1)//2
%o Sm = SymmetricGroup(m)
%o denom = factorial(p)
%o elements = []
%o for perm in SymmetricGroup(p):
%o t = perm.tuple()
%o eperm = []
%o for v2 in range(p):
%o for v1 in range(v2):
%o w1, w2 = sorted([t[v1], t[v2]])
%o eperm.append((w2-1)*(w2-2)//2+w1)
%o elements.append(make_permgroup_element(Sm, eperm))
%o for q in range(m//2+1):
%o char = SymmetricGroupRepresentation([m-q, q]).to_character()
%o numer = sum(char(e) for e in elements)
%o print((p, q), numer//denom)
%o # _Andrey Zabolotskiy_, Aug 28 2018
%Y Cf. A005368, A000088, A011848. Length of row n is A039823(n-1).
%K nonn,tabf
%O 0,13
%A _N. J. A. Sloane_, Jul 12 2015
%E 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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