A334279 Irregular table read by rows: T(n, k) is the coefficient of x^k in the chromatic polynomial of the 1-skeleton of the n-dimensional cross polytope, 0 <= k <= 2n.

0, 0, 1, 0, -3, 6, -4, 1, 0, -64, 154, -137, 58, -12, 1, 0, -2790, 7467, -7852, 4300, -1346, 244, -24, 1, 0, -205056, 593016, -698250, 448015, -175004, 43608, -6990, 700, -40, 1, 0, -22852200, 70164670, -89812001, 64407806, -29113410, 8790285, -1822164, 260868, -25405, 1610, -60, 1

Offset

1,5

Comments

A033815 is the number of acyclic orientations of the n-dimensional cross polytope, which is the absolute value of the chromatic polynomial evaluated at -1.

Sums of absolute values of entries in each row give A033815.

These graphs are chromatically unique, that is, there is no nonisomorphic graph with the same chromatic polynomial.

Conjectures from Peter Kagey, Apr 26 2020: (Start)

T(n,1) = -A161131(2n-1).

T(n,2n-2) = A212689(2n - 1).

T(n,2n-1) = A046092(n-1). (End)

Links

Peter Kagey, Table of n, a(n) for n = 1..2600 (first 50 rows)

Chong-Yun Chao and George A. Novacky Jr., On maximally saturated graphs, Discrete Math., 41 (1982), 139-143.

Eric Weisstein's World of Mathematics, Chromatic Polynomial

Eric Weisstein's World of Mathematics, Cocktail Party Graph

Wikipedia, Cross-polytope

Wikipedia, Turán graph

Example

Table begins:
n/k| 0       1      2       3      4       5     6     7   8   9 10
---+---------------------------------------------------------------
  1| 0       0      1
  2| 0      -3      6      -4      1
  3| 0     -64    154    -137     58     -12     1
  4| 0   -2790   7467   -7852   4300   -1346   244   -24   1
  5| 0 -205056 593016 -698250 448015 -175004 43608 -6990 700 -40  1

Crossrefs

A334278 is analogous for the n-dimensional hypercube.

Cf. A033815, A115400.

Cf. A161131, A212689, A046092.

Sequence in context: A288092 A127574 A356501 * A334278 A169842 A185588

Adjacent sequences: A334276 A334277 A334278 * A334280 A334281 A334282

Keyword

sign,tabf,look

Author

Peter Kagey, Apr 21 2020

Status

approved