The equity section of the balance sheet for an s-corporation is the same as the equity section for a regular C corporation. This is because the S-Corporation designation is a tax rather than accounting issue. All S-Corporations have to start out as c corporations. First, you file whatever paperwork (usually a corporate charter or articles of incorporation) your Secretary of State needs to recognize your corporation. After you get notification from the Secretary of State that your paperwork is a-ok, a business can opt to be taxed as an S-Corporation.
You do this by filling out Form 2553 with the Internal Revenue Service. However, nothing about making the election changes the corporation's equity accounts. You'll still have retained earnings and additional paid-in capital.
Next up - the equity section of the balance sheet for a partnership.