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).

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

Author

N. J. A. Sloane, Jul 12 2015

Keywords

• nonn
• tabf

Comments

See Merris and Watkins (1983) for precise definition.

Examples

			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
...
		

Links

• 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).

Crossrefs

Cf. A005368, A000088, A011848. Length of row n is A039823(n-1).

Row n is apparently formed by the first differences of the first half of row n of A008406.

Programs

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)
# Andrey Zabolotskiy, Aug 28 2018

Formula

From Andrey Zabolotskiy, Aug 28 2018: (Start)

Sum_{ k=0..A011848(n) } T(n,k) * (n*(n-1)/2 - 2*k + 1) = A000088(n).

T(n,k) = A005368(k) for n >= 2*k. (End)

Extensions

Name edited, terms T(7, 9)-T(7, 10) and rows 0-2, 8, 9 added by Andrey Zabolotskiy, Sep 06 2018