A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.

1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -5, 10, -9, 3, 0, 1, -9, 36, -79, 98, -64, 17, 0, 1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0, 1, -28, 378, -3242, 19648, -88676, 306308, -819933, 1703404, -2723374, 3285552, -2887734, 1739326, -639065, 107435, 0

Offset: 1

Author

Alois P. Heinz, Jun 15 2012

Keywords

• sign
• tabf

Comments

The ranked poset L(n) of partitions is defined in A002846. A partition of n into k parts is connected to another partition of n into k+1 parts that results from splitting one part of the first partition into two parts.

Examples

			L(5):     (32)---(221)
         /    \ /     \
        /      X       \
       /      / \       \
    (5)---(41)---(311)---(2111)---(11111)
Chromatic polynomial: q^7-9*q^6+36*q^5-79*q^4+98*q^3-64*q^2+17*q.
Triangle T(n,k) begins:
  1,   0;
  1,  -1,   0;
  1,  -2,   1,    0;
  1,  -5,  10,   -9,    3,     0;
  1,  -9,  36,  -79,   98,   -64,   17,     0;
  1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0;
		

Links

• Alois P. Heinz, Rows n = 1..9, flattened
• Olivier Gérard, The ranked posets L(2),...,L(8)
• Eric Weisstein's World of Mathematics, Chromatic Polynomial
• Wikipedia, Chromatic Polynomial

Crossrefs

Row lengths give: 1+A000041(n) = A052810(n).

Row sums (for n>1) and last elements of rows give: A000004.

Columns k=1-2 give: A000012, (-1)*A000097(n-2).

Cf. A002846, A213242, A213385, A213427.

Extensions

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016