Counting Ordered Permutation Tuples with Lexicographical and Inversion Constraints

Given three positive integers n, m, and mod.

For each positive integer i (1 ≤ i ≤ n), consider all i-element permutations of {1,2,…,i}. Let these permutations be listed in lexicographical order (i.e. dictionary order). For each permutation, define its inversion count as the number of pairs (p, q) with 1 ≤ p < q ≤ i such that the element at position p is greater than the element at position q.

Now, for a fixed i and an integer j (1 ≤ j ≤ m), let f(i,j) be the number (modulo mod) of ordered j-tuples of distinct permutations (p₁, p₂, …, p_j) chosen from the set of all i-permutations, satisfying:

• Lexicographical condition: p₁ < p₂ < … < p_j (when compared as sequences in dictionary order).
• Inversion count condition: inv(p₁) > inv(p₂) > … > inv(p_j), where inv(p) denotes the inversion count of permutation p.

You are to output the table of values f(i,j) modulo mod for all 1 ≤ i ≤ n and 1 ≤ j ≤ m. For each i from 1 to n, output one line containing m space‐separated integers: f(i,1), f(i,2), …, f(i,m). If no valid j-tuple exists then f(i,j) is 0.

Note: The permutations for each i are considered in lexicographical order. A valid j-tuple is equivalent to choosing a subset of j permutations (ordered increasingly by their lex order) for which their inversion counts form a strictly decreasing sequence.

inputFormat

The input consists of a single line with three integers n, m, and mod (1 ≤ n, m; mod ≥ 1).

outputFormat

Output n lines. The i-th line (1-indexed) should contain m integers separated by a space: f(i,1), f(i,2), …, f(i,m) computed modulo mod.

sample

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