Fair Value Measurements (Details) - USD ($) $ in Thousands	12 Months Ended
	Sep. 30, 2022	Jan. 03, 2022	Sep. 30, 2021	Sep. 30, 2020
Deferred compensation plan assets
Mutual Funds			$ 17,806
Liabilities:
Unrealized gain	$ (2,537)
Decrease in fair value of earn-out liability	$ 150
Frontier
Liabilities:
Percentage of voting interest remaining	32.00%
Liabilities, fair value adjustment	$ 2,091
Mar Cor
Deferred compensation plan assets
Earn-out assets related to acquisitions	11,597	$ 7,824	0
Liabilities:
Increase in the fair value of the Earn Out asset	3,773
Fair Value, Inputs, Level 1 | Fair Value, Recurring
Deferred compensation plan assets
Cash	902		1,251
Mutual Funds	12,330
Fair Value, Inputs, Level 1 | Cash | Fair Value, Recurring
Assets:
Fair value of plan assets	40		831
Fair Value, Inputs, Level 2 | Fair Value, Recurring
Assets:
Interest Rate Derivative Assets, at Fair Value	49,952		3,127
Deferred compensation plan assets
Foreign currency forward contracts	507		24
Liabilities:
Pension plan	(26,654)		(46,013)
Deferred compensation plan liabilities	(20,081)		(24,382)
Total return swaps—deferred compensation	(632)		(130)
Long‑term debt	(884,517)		(752,988)
Interest rate swaps			(303)
Foreign currency forward contracts	(872)		(102)
Fair Value, Inputs, Level 2 | Commodity Contract | Fair Value, Recurring
Liabilities:
Derivative Liability	(7)		(19)
Fair Value, Inputs, Level 3 | Fair Value, Recurring
Liabilities:
Earn-out liabilities related to acquisitions	0		(150)	$ (295)
Purchase Right	0		8,305
Fair Value, Inputs, Level 3 | Fair Value, Recurring | Other Current Liabilities
Liabilities:
Earn-out liabilities related to acquisitions	0		(150)	(295)
Fair Value, Inputs, Level 3 | Fair Value, Recurring | Other Noncurrent Liabilities
Liabilities:
Earn-out liabilities related to acquisitions	0		0	$ 0
Net Asset Value | Global Multi-Asset Fund | Fair Value, Recurring
Assets:
Plan assets at net asset value	11,632		15,244
Net Asset Value | Government Securities | Fair Value, Recurring
Assets:
Plan assets at net asset value	3,343		5,158
Net Asset Value | Liability Driven Investment | Fair Value, Recurring
Assets:
Plan assets at net asset value	928		2,793
Net Asset Value | Guernsey Unit Trust | Fair Value, Recurring
Assets:
Plan assets at net asset value	2,048		2,387
Net Asset Value | Global Absolute Return | Fair Value, Recurring
Assets:
Plan assets at net asset value	$ 1,299		$ 2,225