OFFSET
1,4
COMMENTS
T(r,g,b) is the number of injectively 3-colored trees with r red vertices, g green vertices, and b blue vertices, including a root vertex which is colored blue.
Summing T(r,g,b) over all r,g,b such that r+g+b=n yields the n-th Catalan number, A000108(n).
Column (or row) sums within each fixed r+g+b=n layer yield the number of ordered trees on n edges containing a fixed number of nodes adjacent to a leaf, A108759(n).
Main antidiagonal (corresponding to maximal value b = ceiling((r+g+b)/2)) within each fixed odd (r+g+b) layer is the number of "fighting fish" with fixed numbers of left lower free and right lower free edges with a marked tail A278880.
LINKS
T. Einolf, R. Muth and J. Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021, Remark 4.7.
FORMULA
T(r,g,b) = (r+g+b)/((g+b)*(r+b))*C(r+g,b-1)*C(g+b,r)*C(r+b,g).
T(r,g,b) = (r+g+b)/((g+b)*(r+b))*(r+g)!/((r+g-b+1)!*(b-1)!)*((g+b)!/(g+b-r)!*r!))*((r+b)!/((r+b-g)!*g!).
EXAMPLE
The first few layers of the pyramidal array are:
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1 (r+g+b=1), (b=1) T(0,0,1)
LAYER SUM: 1
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1 1 (r+g+b=2), (b=1) T(0,1,1) T(1,0,1)
LAYER SUM: 2
-----------------------------------------------------------------------
3 (r+g+b=3), (b=1) T(1,1,1)
1 1 (r+g+b=3), (b=2) T(0,1,2) T(1,0,2)
LAYER SUM: 5
-----------------------------------------------------------------------
2 2 (r+g+b=4), (b=1) T(1,2,1) T(2,1,1)
1 8 1 (r+g+b=4), (b=2) T(0,2,2) T(1,1,2) T(2,0,2)
LAYER SUM: 14
-----------------------------------------------------------------------
5 (r+g+b=5), (b=1) T(2,2,1)
15 15 (r+g+b=5), (b=2) T(1,2,2) T(2,1,2)
1 5 1 (r+g+b=5), (b=3) T(0,2,3) T(1,1,3) T(2,0,3)
LAYER SUM: 42
-----------------------------------------------------------------------
3 3 (r+g+b=6), (b=1) T(2,3,1) T(3,2,1)
8 54 8 (r+g+b=6), (b=2) T(1,3,2) T(2,2,2) T(3,1,2)
1 27 27 1 (r+g+b=6), (b=3) T(0,3,3) T(1,2,3) T(2,1,3) T(3,0,3)
LAYER SUM: 132
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7 (r+g+b=7), (b=1) T(3,3,1)
70 70 (r+g+b=7), (b=2) T(2,3,2) T(3,2,2)
42 168 42 (r+g+b=7), (b=3) T(1,3,3) T(2,2,3) T(3,1,3)
1 14 14 1 (r+g+b=7), (b=4) T(0,3,4) T(1,2,4) T(2,1,4) T(3,0,4)
LAYER SUM: 429
-----------------------------------------------------------------------
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Robert Muth, Aug 05 2023
STATUS
approved