The national debt and currency are financial assets of the private sector and local governments. Since currency can be withdrawn on demand from the aggregate bank sector the ratio of national debt to fiat currency is determined as a portfolio choice of the private sector and local governments.
The central bank and/or commercial bank sector must use its balance sheet to allow the private sector and local governments to swap currency for other financial assets otherwise bank deposits would not keep par with currency. Some of these financial asset swaps are swaps of government securities for currency or currency for government securities.
FRED graph Federal Debt Held by Federal Reserve Banks:
Notice the smooth increase up until Q3 2007 when Fed began to lend funds to the bank sector at the discount window, which adds reserves to the aggregate bank, and Fed sold Treasury securities to drain reserves so it could keep control over the federal funds rate. This strategy would not work in the long run because Fed would run out of Treasuries and then have no way to neutralize reserves added by discount window loans if the volume of lending became larger than Treasury assets.
FRED graph Currency in Circulation:
If you convert these two graphs to the same time on the horizontal axis and the same units (millions or billions) on the vertical axis then it shows Fed holding Treasuries roughly equal to the currency in circulation up until Q3 2007 as discussed above. This means when the aggregate private sector and local governments would withdraw currency from the aggregate bank the Fed would go into the open market and purchase Treasuries to offset or "service" the currency drain. This was necessary when Fed kept the pool of bank reserves as small as possible to control the federal funds rate. Banks give reserves to Fed to buy vault cash (currency) and would not have any reserves if Fed did not provide those reserves while servicing the currency drain.
Banks must allow depositors to withdraw currency to keep checking deposits on par with currency since both currency and deposits are used to clear payment. Fed must service the currency drain during some operating conditions to retain control of the federal funds rate or also known as the monetary policy target rate. When Fed would buy Treasuries to service the currency drain this meant that the private sector and local governments had the option to hold either Treasury securities or currency in their portfolio of assets consisting of currency and Treasuries.
After Q3 2007 Fed sold and then purchased Treasuries without matching holdings to its currency liabilities so the currency holdings would be at the election of the private sector and local governments demand for currency via withdrawal from the aggregate bank. The distribution of Treasuries between Fed and the private sector and local governments would be in the discretion of Fed under efforts to manage monetary policy using its balance sheet.
So in general the amount of Treasury debt outside government, held by Fed, private sector, and local governments, equals the sum of net deficit spending over the history of the nation. The currency outside banks is determined by the portfolio choice of the private sector and local governments. The ratio of currency to Treasuries does not appear to be a factor correlated with inflation. Total deficit spending by households and firms is related to aggregate credit (loans). This combines with deficit spending by governments and a shortage of resources to drive inflation.
I published three papers on this subject matter which draw conclusions from research in the United States flow of funds and other sources.
Sources and Sinks of the M1 Money in a Four Sector Model of the U.S. Financial System.
Currency drain analysis from 1980-2011 is on pages 11-13 including Figures 4 and 5 which relate to the answer above.