Exhibit 32.1

CERTIFICATION PURSUANT TO 18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

The certification set forth below is being submitted in connection with the Annual Report of Publix Super Markets, Inc. (the “Company”) on Form 10-K for the period ending December 29, 2012 (the “Report”) pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002.

I, William E. Crenshaw, Chief Executive Officer of the Company, certify, to the best of my knowledge, that on the date hereof:

(1) The Report fully complies with the requirements of Section 13(a) or 15(d) of the Exchange Act; and

(2) The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

/s/ William E. Crenshaw

William E. Crenshaw

Chief Executive Officer

February 28, 2013