OFFSET
0,4
COMMENTS
If g is taken as g(X) = X + g_2*X^2 + g_3*X^3 + ... then the compositions are (g circle f)(X) = g(f(X)) = X and (f circle g)(X) = f(g(X)) = X.
Lexicographically descending in the rows, i.e., f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4).
This is another version of A111785, where each row is sorted lexicographically ascending, i.e., f(1)^4 f(4)^2 (+4) < f(1)^3 f(2)^2 f(5) (-36).
REFERENCES
Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 1953.
FORMULA
g(n) := f(1)^(-n) Sum_{j(2), j(3), ...} (-1)^{j(2) + j(3) + ...} ((n-1 + j(2) + j(3) + ...)!)/(n! j(2)! j(3)! ...) ((f(2))/(f(1))^j(2) ((f(3))/(f(1)))^j(3) ...
The sum is to be taken over all combinations of the exponents {j(2), j(3), j(4), ...} with j(2) + 2j(3) + 3j(4) + ... = n-1. See Morse, P. M. and Feshbach, H. pp. 411-413.
EXAMPLE
Matrix lexicographically descending in the rows:
for instance f(5) f(2)^2 f(1)^3 (-36) > f(4)^2 f(1)^4 (+4)
1;
-1;
-1,2;
-1,5,-5;
-1,6,3,-21,14;
-1,7,7,-28,-28,84,-42;
-1,8,8,-36,4,-72,120,-12,180,-330,132;
-1,9,9,-45,9,-90,165,-45,-45,495,-495,165,-990,1287,-429;
-1,10,10,-55,10,-110,220,5,-110,-55,660,-715,-55,330,660,-2860,2002,55,-1430,5005,-5005,1430;
PROG
(MuPAD)
alfa:=["a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k"]:
byRow := proc(od, // original weighted degree
wd, // remaining weighted degree
il, // index of last indeterminate
jl, // exponent of last indeterminate
ni, // remaining number of indeterminates
lx) // lexicographic string
local j;
begin
if wd > 1 then
for j from min(wd, il) downto 2 do:
if j >= il then
j:=il: // stay at the latest indeterminate
byRow(od, wd-j+1, j, jl+1, ni-1, lx.alfa[j]):
else // advance to next indeterminate
byRow(od, wd-j+1, j, 1 , ni-1, lx.alfa[j]):
end_if:
end_for:
else // output the monomial
dd:=1: d0:="+": dc:=1:
for j from length(lx)-1 downto 0 do:
d1:=substring(lx, j):
if d1 <> d0 then
d0:=d1: dc:=1: dd:=-dd:
else // the indeterminate changes
dc:=dc+1: dd:=-dd*dc:
end_if:
end_for:
nn:=fact(2*od-ni-2)/fact(od): // rising factorial
// One row of A304462: coefficients of the lexicographically descending monomials:
print(nn/dd):
// One row of A304462: coefficients of the lexicographically descending monomials
// plus some representation of the monomials themselves:
// for j from 1 to ni do:
// lx:=lx."a":
// end_for:
// print(nn/dd, lx): // monomial lx
end_if:
end_proc:
// Output the 8th row:
n:=8:
byRow(n, n, n, 0, n-1, "")
CROSSREFS
KEYWORD
tabf,sign
AUTHOR
Herbert Eberle, May 13 2018
STATUS
approved