cp's OEIS Frontend

if i = 1: a(n,k,i) = sum( a(n-1,k,j) for j in [1..k] )

if i = k: a(n,k,i) = sum( a(n-1,k-1,j) + a(n-1,k,j) for j in [1..k-1] )

otherwise: a(n,k,i) = sum( a(n-1,k,j) for j in [1..i-1] )

where n is the length, k is the number of right to left minima and i is the position of the maximum in relation to the right to left minima.

Initial Conditions: if k > n or i > k then a(n,k,i) = 0, if k = 1 then a(n,k,i) = 1.

Then a(n) = sum( sum( a(n,k,i) for i in [1..k]) for k in [1..n] ).

G.f: 1 + x * sum(x^n * F_n(1+x) for n >= 0) where F_n(q) = sum( [n,m] for m in [0..n] ). Note [n,m] is the q-binomial. - Christian Bean, Jun 03 2015

G.f: 1 + x/(1-x) * sum( sum( x^(i+k) * F_n+1,k(1/(1-x)) for k in [0..n+1] ) for n >= 0 ) where F_n,k(q) = q^n * q^C(k,2) * [n-1,k-1] and [a,b] is the q-binomial. - Christian Bean, Jun 03 2015

If x appears after x-1 in the permutation then we say that x is a ceiling point.

if i = 1: aup(n,k,i,l) = sum( abar(n,k,i,l) for m in [0..k] )

otherwise: aup(n,k,i,l) = sum( abar(n-1,k,1,m) for m in [l..k] ) + sum( sum( adown(n-1,k,j,m) for m in [i..k]) for j in [1..i-1] )

abar(n,k,i,l) = sum( a(n-1,k-1,j,l-1) for j in [1..k-1] )

adown(n,k,i,l) = sum( aup(n-1,k,j,l) + adown(n-1,k,j,l) for j in [i..k] )

a(n,k,i,l) = aup(n,k,i,l) + adown(n,k,i,l) + abar(n,k,i,l)

where n is the length, k is the number of left to right minima, i is the position of the maximum, l is the position of the first ceiling point

aup implies that max is a ceiling point, abar implies that max is a left to right minimum and adown implies max is neither.

Initial conditions: if i > l or k > n or i > k or l > k then aup(n,k,i,l) = adown(n,k,i,l) = 0, if i < l or l <= 0 then aup(n,k,i,l) = 0, if n - k = 1 then a(n,k,i,l) = 1, if i does not equal 1 the abar(n,k,i,l) = 0, abar(n,n,1,0) = 1.

a(n) = sum( sum( sum( a(n,k,j,m) for m in [0..k] ) for j in [1..k] ) for k in [1..n] )