cp's OEIS Frontend

Define the matrix function matexps(M) to be exp(M)/exp(1). Then the number of k-level labeled rooted trees with n leaves is also column 0 of the triangle resulting from the n-th iteration of matexps on the Pascal matrix P, A007318. The resulting triangle is also S^n*P*S^-n, where S is the Stirling2 matrix A048993. This function can be coded in PARI as sum(k=0,200,1./k!*M^k)/exp(1), using exp(M) does not work. See A056857, which equals (1/e)*exp(P) or S*P*S^-1. - Gerald McGarvey, Aug 19 2009

For any pair (X,Y) meeting the requirement, so does the pair (Y,X) which must be distinct from (X,Y), except for X = Y = {} when n = 0. Therefore all a(n) are even for n > 0. - M. F. Hasler, Dec 30 2020

Stirling2 transform of A000587. 2nd Stirling2 transform of A033999. - Vladimir Reshetnikov, Oct 22 2015

First differs from A046682 at a(11) = 28, A046682(11) = 29.

A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:

1 2 3

1 2

2 3

Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

The triangle reflects the Jordan-decomposition of the matrix of Stirling numbers of the second kind. A display of the matrix formula can be found at the Helms link which also explains the generation rule for the A()-numbers in a different way. - Gottfried Helms Apr 19 2014

From Gus Wiseman, Jan 02 2020: (Start)

Also the number of balanced reduced multisystems with atoms {1..n} and depth k. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. For example, row n = 4 counts the following multisystems:

{1,2,3,4} {{1},{2,3,4}} {{{1}},{{2},{3,4}}}

{{1,2},{3,4}} {{{1},{2}},{{3,4}}}

{{1,2,3},{4}} {{{1},{2,3}},{{4}}}

{{1,2,4},{3}} {{{1,2}},{{3},{4}}}

{{1,3},{2,4}} {{{1,2},{3}},{{4}}}

{{1,3,4},{2}} {{{1},{2,4}},{{3}}}

{{1,4},{2,3}} {{{1,2},{4}},{{3}}}

{{1},{2},{3,4}} {{{1}},{{3},{2,4}}}

{{1},{2,3},{4}} {{{1},{3}},{{2,4}}}

{{1,2},{3},{4}} {{{1,3}},{{2},{4}}}

{{1},{2,4},{3}} {{{1,3},{2}},{{4}}}

{{1,3},{2},{4}} {{{1},{3,4}},{{2}}}

{{1,4},{2},{3}} {{{1,3},{4}},{{2}}}

{{{1}},{{4},{2,3}}}

{{{1},{4}},{{2,3}}}

{{{1,4}},{{2},{3}}}

{{{1,4},{2}},{{3}}}

{{{1,4},{3}},{{2}}}

(End)

From Harry Richman, Mar 30 2023: (Start)

Equivalently, T(n,k) is the number of length-k chains from minimum to maximum in the lattice of set partitions of {1..n} ordered by refinement. For example, row n = 4 counts the following chains, leaving out the minimum {1|2|3|4} and maximum {1234}:

(empty) {12|3|4} {12|3|4} < {123|4}

{13|2|4} {12|3|4} < {124|3}

{14|2|3} {12|3|4} < {12|34}

{1|23|4} {13|2|4} < {123|4}

{1|24|3} {13|2|4} < {134|2}

{1|2|34} {13|2|4} < {13|24}

{123|4} {14|2|3} < {124|3}

{124|3} {14|2|3} < {134|2}

{134|2} {14|2|3} < {14|23}

{1|234} {1|23|4} < {123|4}

{12|34} {1|23|4} < {1|234}

{13|24} {1|23|4} < {14|23}

{14|23} {1|24|3} < {124|3}

{1|24|3} < {1|234}

{1|24|3} < {13|24}

{1|2|34} < {134|2}

{1|2|34} < {1|234}

{1|2|34} < {12|34}

(End)

Also the number of cells of dimension k in the fine subdivision of the Bergman complex of the complete graph on n vertices. - Harry Richman, Mar 30 2023

A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:

1 2 3

1 2

2 3