Let 〈a1, a2, . . . , an〉 be a sequence of comparable elements. In this paper, we study two constrained versions of the longest increasing subsequence (LIS) problem. The first problem is the range-constrained longest increasing subsequence (RLIS) problem. Given 0 < LI ≤ UI < n and 0 ≤ LV ≤ UV , the objective of the RLIS problem is to deliver a maximum-length increasing subsequence 〈ai1 , ai2 , ....

The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n−1 2 th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous resul...

where Rk,r is the number of permutations of the integers {1, . . . , k} that contain an increasing subsequence of length at least r. Let Xy denote a positive integer valued random variable such that Prob (Xy ≥ r) = F (r; y). If R k,r denotes the complement of Rk,r, i.e. the number of permutations σ ∈ Sk all of whose increasing subsequences have length strictly less than r, then clearly R k,r = ...

Watermelon ( Citrulluslanatus ) which originated in Africa is widely cultivated throughout the world including India. This fruit known for its nutritional value and relished by many people. Only pulp of this used commercially rind also rich so nutrients considered agricultural waste. To add economic to nutritionally waste i.e., watermelon rind, three edible value-added products were prepared ev...