{"problem":{"name":"Sorting Game","description":{"content":"Takahashi and Snuke came up with a game that uses a number sequence, as follows: *   Prepare a sequence of length $M$ consisting of integers between $0$ and $2^N-1$ (inclusive): $a = a_1, a_2, \\ldots","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":3000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"nomura2020_f"},"statements":[{"statement_type":"Markdown","content":"Takahashi and Snuke came up with a game that uses a number sequence, as follows:\n\n*   Prepare a sequence of length $M$ consisting of integers between $0$ and $2^N-1$ (inclusive): $a = a_1, a_2, \\ldots, a_M$.\n    \n*   Snuke first does the operation below as many times as he likes:\n    *   Choose a positive integer $d$, and for each $i$ $(1 \\leq i \\leq M)$, in binary, set the $d$\\-th least significant bit of $a_i$ to $0$. (Here the least significant bit is considered the $1$\\-st least significant bit.)\n*   After Snuke finishes doing operations, Takahashi tries to sort $a$ in ascending order by doing the operation below some number of times. Here $a$ is said to be in ascending order when $a_i \\leq a_{i + 1}$ for all $i$ $(1 \\leq i \\leq M - 1)$.\n    *   Choose two adjacent elements of $a$: $a_i$ and $a_{i + 1}$. If, in binary, these numbers differ in exactly one bit, swap $a_i$ and $a_{i + 1}$.\n\nThere are $2^{NM}$ different sequences of length $M$ consisting of integers between $0$ and $2^N-1$ that can be used in the game.\nHow many among them have the following property: if used in the game, there is always a way for Takahashi to sort the sequence in ascending order regardless of Snuke's operations? Find the count modulo $(10^9 + 7)$.\n\n## Constraints\n\n*   All values in input are integers.\n*   $1 \\leq N \\leq 5000$\n*   $2 \\leq M \\leq 5000$\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $M$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"nomura2020_f","tags":[],"sample_group":[["2 5","352\n\nConsider the case $a = 1, 3, 1, 3, 1$ for example.\n\n*   When the least significant bit of each element of $a$ is set to $0$, $a = 0, 2, 0, 2, 0$;\n*   When the second least significant bit of each element of $a$ is set to $0$, $a = 1, 1, 1, 1, 1$;\n*   When the least two significant bits of each element of $a$ are set to $0$, $a = 0, 0, 0, 0, 0$.\n\nIn all of the cases above and the case when Snuke does no operation to change $a$, we can sort the sequence by repeatedly swapping adjacent elements that differ in exactly one bit. Thus, this sequence has the property: if used in the game, there is always a way for Takahashi to sort the sequence in ascending order regardless of Snuke's operations."],["2020 530","823277409"]],"created_at":"2026-03-03 11:01:14"}}