Morton Abramson

Generalizations of Terquem's problem

Abstract Terquems problem asks for the number of combinations 1≤ x 1 2 p ≤n with even entries in even position and odd entries in odd position. This is generalized to x 1 ≡1+ k 1 (mod m ), x j ≡x j-1 +1+ k j (mod m ), j =2(1) p . The case m =2, k j =0, j =1(1) p is Terquems problem, while k j =0, j =1(1) p is Skolems generalization. Also considered is the “circular” type problem which results when the condition x 1 ≡ 1+k 1 (mod m ) is deleted.

Enumeration of combinations with restricted differences and cospan

Abstract For the p -combination 1≤ x 1 x 2 x p ≤ n the differences are d j =x j+1 −x j , j =1(1) p −1, the span is d=x p −x 1 , and the cospan is n−d . An explicit expression (14) is obtained for the number of p -combinations satisfying the conditions k≤d j ≤k′ , j =1(1) p −1, l≤n−d≤l′ . Recurrence relations are found. Special cases include generalized Fibonacci and Lucas numbers.

Enumeration of arrays by column rises

in which each row consists of a permutation of 1, 2,..., n, denote the ith column by C. Write Ci < Ci+1 if ei < ei+, , fi < fi+l ,..., xi < x~+~ and call such an event a column rise. Denote by R(m, H, l) the number of arrays (1) containing precisely f column rises. The number R(1, n, t) is the well known Eulerian number [l] and the enumeration for the case m = 2 was announced in [2, Theorem 21 and proved in [3, Theorem 2.11. In Section I, by purely combinatorial methods, we find a formula and generating function for R(m, n, f). In Section 2 we consider some estimations and the behaviour of R(tr, n, 0) for large n.