Re: combinatorial interpretation of generalised berneuler numbers


i had wanted to post this as a part of something else
but events have conspired against me
and i think this may still be interesting practice
ce to some

this is one of those challenges
that in one way doesn't seem to involve math at all
and yet is one of those skills i think is at the
heart of mathematics

the challenge:
_interpret_ the following theorem as a
a combinatorial identity

-----------------------------
define:
the generalised (exponential-form) berneuler
er numbers are

m oo
x --- m
------ = \ gb
|m x / n j j
| e --- ---- x
|n j=0 (1)
j

for n=2
these cover the classical
euler and bernoulli numbers

the identity these berneulers obey is:

m m
--- (1) gb gb
m \ p+1 n j n k
(p + 1 - m) gb = - / -------------------
n p+1 --- (1) (1) (1)
j+k <= p j k p-j-k
p-j-k = m-1 (mod n)

darndarndarn!!

everytime i try one of these challenges
i always make a stupid typo along the way

the multiplier is (p+1-2m)
and the the berneuler is m^n_gb_(p+1-m)

the above came from the wrong page
where i had made an error down the line

anyways
the rest remains

the challenge is to interpret this
as a combinatorial statement
on some data structure

below i give two hints
don't scroll if don't want spoilers

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hint1

i won't prove this identity here
but instead just sketch:
- differentiate in two different ways
- one of the ways involves a heroic use of the
he chain rule
- the other just differentiates a series

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hint2

there is a great book by goulden and jackson
"combinatorial enumeration"
which explains the coolest language
or "four laws" of combinatorial interpretation
on which to interpret many classic sums
(dover has recently picked it up)

the above derivation relies on differentiation
of something that is a reciprocal
of a series that has a simple (exponential)
l) interpretation
(it's coefficient is 1 if position = m(mod n)
else 0)
so...

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hint3

okay
so i lied about two hints

also remember
there is an interpretation
of classical euler and bernoulli numbers
in alternating permutations

this provides a starting data structure to look at
but i think it still doesn't give it away

what type of strange walks are needed for higher n?


-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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