Given a binary word [omega] of length N the number of zeroes in [omega] between the (j - 1)-st and the j-th one in [omega], where the ones are counted from left to right, is denoted by [[absolute value of [omega]].

A quadruple ([alpha], [beta]; [gamma], [delta]) is said to be the boundary of a rhomboid-shaped Knutson-Tao-puzzle P if [alpha] equals the binary word that is obtained by reading the labels along the leftmost NE-SW diagonal of P from left to right, [beta] equals the word that is obtained by reading the labels along the rightmost NW-SE diagonal of P from left to right, [gamma] equals the word that is obtained by reading the labels along the rightmost NE-SW diagonal of P from left to right and [delta] the binary word that is obtained by reading the labels along the leftmost NW-SE diagonal of P from left to right.

One of these conditions states that d(w) [greater than or equal to] d(u) + d(v) where u, v, w are binary words that encode the boundary conditions of a TFPL and d(u) denotes the number of inversions of u.

Recall that the descents of an ordinary composition are the positions of the ones in the associated binary word.

K) are the compositions whose binary words are w (resp.

More precisely, for a finite or infinite binary word x = [x.

In the above setting, the problem is of finding, for any infinite binary word x, relations between the quantities

More precisely, for each of the lower and the upper bounds, we construct a concrete example of an infinite binary word that attains the equality in the bound.

Moreover, all the bounds are best possible except trivial exceptions, in the following sense: For any 2/5 [less than or equal to] r [less than or equal to] 1, there exists an infinite binary word x such that [r.

In this subsection, we give an infinite binary word x for any 2/5 [less than or equal to] r [less than or equal to] 1 such that [absolute value of I([x.

l], we replace the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with a new sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of infinite

binary words, close to the [t.

nn], then, count zigzag-free

binary words with are equal number of zeros and ones.