A321931 -id:A321931 - cp's OEIS frontend

A321979 Number of e-positive simple labeled graphs on n vertices.

1, 1, 2, 8, 60, 899

Offset: 0

Gus Wiseman, Nov 23 2018

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895). A graph is e-positive if, in the expansion of its chromatic symmetric function in terms of elementary symmetric functions, all coefficients are nonnegative.

			The 4 non-e-positive simple labeled graphs on 4 vertices are:
  {{1,2},{1,3},{1,4}}
  {{1,2},{2,3},{2,4}}
  {{1,3},{2,3},{3,4}}
  {{1,4},{2,4},{3,4}}

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, The a(4) = 60 e-positive simple labeled graphs.

Cf. A000569, A006125, A229048, A240936, A277203, A321895, A321911, A321918, A321914, A321931, A321980, A321981, A321982.

A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 2, 3, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, -2, -1, -2, -1, 0, 6, 3, 8, 6, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 0, -6, -6, -5, -3, -3, -1, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1
  (11):  1  1
.
  (3):    1
  (21):  -1 -1
  (111):  2  3  1
.
  (4):    -1
  (22):    1  1
  (31):    1     1
  (211):  -2 -1 -2 -1
  (1111):  6  3  8  6  1
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    2  1  2  1
  (311):    2  2  1     1
  (2111):  -6 -6 -5 -3 -3 -1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: F(32) = -p(5) - p(32).

Wikipedia, Symmetric polynomial

Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A322012 Number of s-positive simple labeled graphs with n vertices.

Original entry on oeis.org

1, 2, 8, 60, 1009

Offset: 1

Gus Wiseman, Nov 24 2018

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is the augmented monomial symmetric function basis (see A321895). A graph is s-positive if, in the expansion of its chromatic symmetric function in terms of Schur functions, all coefficients are nonnegative.

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.

a(n) >= A321979(n).

Cf. A000569, A006125, A229048, A240936, A277203, A321895, A321924, A321925, A321931, A321994.

A322066 Number of e-positive antichains of sets spanning n vertices.

Original entry on oeis.org

1, 1, 2, 8, 64, 1299

Offset: 0

Gus Wiseman, Nov 25 2018

A stable partition of a hypergraph or set system is a set partition of the vertices where no non-singleton edge has all its vertices in the same block. The chromatic symmetric function is given by X_G = Sum_pi m(t(pi)) where the sum is over all stable partitions pi of G, t(pi) is the integer partition whose parts are the block-sizes of pi, and m is the basis of augmented monomial symmetric functions (see A321895). A hypergraph or set system is e-positive if, in the expansion of its chromatic symmetric function in terms of elementary functions, all coefficients are nonnegative.

			The a(3) = 8 e-positive antichains:
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
The antichain {{1,2,3}} is not e-positive, as its chromatic symmetric function is -3e(3) + 3e(21).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.

Cf. A006125, A229048, A240936, A277203, A321895, A321914, A321918, A321931, A321979, A321980, A321981, A321982, A321994, A322012.

cp's OEIS Frontend

A321979 Number of e-positive simple labeled graphs on n vertices.

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A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

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A322012 Number of s-positive simple labeled graphs with n vertices.

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A322066 Number of e-positive antichains of sets spanning n vertices.

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