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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).

%I A259976 #33 Jan 26 2025 20:31:05
%S A259976 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 A259976 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 A259976 85,197,426,866,1615,2743,4125,5495,6318,6054,4416,1637
%N 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).
%C A259976 See Merris and Watkins (1983) for precise definition.
%H A259976 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 A259976 Andrey Zabolotskiy, <a href="https://github.com/colt-browning/a259976">a259976</a> (implementation in Rust).
%F A259976 From _Andrey Zabolotskiy_, Aug 28 2018: (Start)
%F A259976 Sum_{ k=0..A011848(n) } T(n,k) * (n*(n-1)/2 - 2*k + 1) = A000088(n).
%F A259976 T(n,k) = A005368(k) for n >= 2*k. (End)
%e A259976 The triangle begins:
%e A259976 [0] 1
%e A259976 [1] 1
%e A259976 [2] 1
%e A259976 [3] 1,0,
%e A259976 [4] 1,0,1,1,
%e A259976 [5] 1,0,1,2,2,0,
%e A259976 [6] 1,0,1,3,4,6,6,3,
%e A259976 [7] 1,0,1,3,5,11,20,24,32,34,17
%e A259976 [8] 1,0,1,3,6,13,32,59,106,181,261,317,332,245,89
%e A259976 [9] 1,0,1,3,6,14,38,85,197,426,866,1615,2743,4125,5495,6318,6054,4416,1637
%e A259976 ...
%o A259976 (Sage)
%o A259976 from sage.groups.perm_gps.permgroup_element import make_permgroup_element
%o A259976 for p in range(8):
%o A259976     m = p*(p-1)//2
%o A259976     Sm = SymmetricGroup(m)
%o A259976     denom = factorial(p)
%o A259976     elements = []
%o A259976     for perm in SymmetricGroup(p):
%o A259976         t = perm.tuple()
%o A259976         eperm = []
%o A259976         for v2 in range(p):
%o A259976             for v1 in range(v2):
%o A259976                 w1, w2 = sorted([t[v1], t[v2]])
%o A259976                 eperm.append((w2-1)*(w2-2)//2+w1)
%o A259976         elements.append(make_permgroup_element(Sm, eperm))
%o A259976     for q in range(m//2+1):
%o A259976         char = SymmetricGroupRepresentation([m-q, q]).to_character()
%o A259976         numer = sum(char(e) for e in elements)
%o A259976         print((p, q), numer//denom)
%o A259976 # _Andrey Zabolotskiy_, Aug 28 2018
%Y A259976 Cf. A005368, A000088, A011848. Length of row n is A039823(n-1).
%Y A259976 Row n is apparently formed by the first differences of the first half of row n of A008406.
%K A259976 nonn,tabf
%O A259976 0,13
%A A259976 _N. J. A. Sloane_, Jul 12 2015
%E A259976 Name edited, terms T(7, 9)-T(7, 10) and rows 0-2, 8, 9 added by _Andrey Zabolotskiy_, Sep 06 2018